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Applied Discrete Mathematics
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Key Concepts:
Terms in this set (387)
Identity Laws
p^T≡p | pVF≡F
Domination Laws
pvT≡T | p^F≡F
Idempotent Laws
pvp≡p | p^p≡p
Double Negatation Laws
!(!p)≡p
Commutative Laws
pvq≡pvp | p^q≡q^p
Associative Laws
(pvq)vr≡pv(qvr) | (p^q)^r≡p^(q^r)
Distributive Laws
pv(q∧r)≡(pvq)∧(pvr) | p^(qvr)≡(p^q)v(p^r)
De Morgan's Laws
!(p^q)≡!pv!q | !(pvq)≡!p^!q
Absorbtion Laws
pv(p^q)≡p | p^(pvq)≡p
Negation Laws
pv!p=T | p^!p=F
Let A = {1,3,5,a} and B = {a,b,c,d,3}. Find A∩B
{a,3}
If we let A = {1,3,4} and B = {1,2,3,4,5,t,h,e} then A⊆B
true
Express the set {-1,0,1,2,3} in set-builder notation.
{x | -2 < x < 4}
Find the cardinality of {a,1,b,c}.
4
Let A = {7,8,9} and B = {Ø}. Find A ∪ B.
{7,8,9,Ø}
Find the bitwise AND of the two bit strings 1110 and 0101.
0100
Let A = {0,1} and B = {r,t,y}. Then
A x B = {(r,0), (r,1),(t,0),(t,1),(y,0)(y,1)}.
False
Let A = {7, {7}, {7,{7}}} and B = {7,{7}}. Is B⊂A?
Yes
{1,3} ∈ {1,3,{1}},{3}}
False
Find |A|, where A ={3,{3},2,{2,3}}.
4
Let U = {a,l,g,o,r,i,t,h,m} and A = {i,t}. Find A^c.
{a,l,g,o,r,h,m}
{1,Ø} ⊂ {{1},Ø,{Ø}}
False
Let A = {g,r,a,y}, B = {t,a,n}, and C = {b,l,u,e}. Find A ∪ B ∪ C.
{g,r,a,y,t,n,b,l,u,e}
Let A = {1,2,3,4,a} and B = {a,n,d,1,2}. Find the following:
(a) A - B
(b) B - A
(a) is {3,4}
(b) is {n,d}
{a,b,c} = {c,a,b,c}
True
Suppose a "for" loop in algorithm takes 2,300 passes for a particular set of input. Subsequent to these passes, it terminates, and then another "for" loop takes 4,500 passes. Then there are 6,800 passes total.
True
Let's say that Sue wants the password on her computer to NOT begin with "Sue." Let's also say that a password must be of length 20. Each character in the password can be any digit 0,1,2,...,9, an uppercase letter, a lowercase letter, and the passwords are case sensitive. Then the number of passwords that meet Sue's requirements can be determined by computing 62^17.
False
The 2-permutations of the set {a,r,t} are ar, ra, at, ta, rt, and tr.
True
Determine the value of C(5,3).
10
The number of ternary strings of length 5 can be determined by computing 5^3.
False
The list of all the 2-combinations of the set {m,a,t,h} is ma, mt, mh, and ah.
False
To determine the number of ways a President, Vice-President, Secretary, Treasurer, and Song Leader for a club can be chosen for a club that has 50 members, one could compute the value of P(50,5).
True
Suppose there are 100 cans of green beans in a grocery store, and 25 of these are "Best Deal" brand green beans. How many ways are there to choose a can of green beans at this grocery store that are not the "Best Deal" brand?
75
Suppose a set S1 has 50 elements and that S2 has 40 elements. There are also 10 elements common to S1 and S2. How many elements are in the union of the sets S1 and S2?
80
Suppose a password must contain 2 characters, where a character can be any of the digits 1,2,3,4,5 or any lowercase letter. Then to compute the number of possible passwords, one would compute 31^2.
True
Suppose that, at a university, the Mathematics Department has two clubs, a Theoretical Math Club, and an Applied Math Club. Let's say that there are 30 Theoretical Math Club members, 20 Applied Math Club members, and 5 members of both clubs. Then there are 50 people that are members of the Theoretical Math Club, Applied Math Club, or both clubs.
False
Suppose a password for an online banking system must have 10 characters, and the characters be any digit 0,1,2,...,9, an uppercase letter, or lowercase letter, and the passwords are case sensitive (meaning that, for example, the password A123456789 is different than the password a123456789). Then, to determine the number of possible passwords, one could compute 62^10.
True
Determine the value of P(3,2).
6
Suppose we have the following scheme for passwords: Each password must be of length 4, 5, or 6. The first character of each password must be a digit 0,1,2,...,9. The remaining characters can be any digits 0,1,2,...,9, an uppercase letter, or a lowercase letter; the passwords are case sensitive. Then, to determine the number of possible passwords, one could compute 62^4 + 62^5 + 62^6.
False
Suppose that a store sells two types of golf balls, namely, brand "Hit it far" and brand "Lots of Spin." There are 15 boxes of "Hit it far" golf balls and 7 boxes of "Lots of Spin" golf balls. How many ways are there to select a box of golf balls?
22
Consider the ceiling function f(a) = ┌ a ┐.
Determine f(-3.5).
-3
Let A = Z and B = Z. Consider the function f:A→B defined by the rule that f(a) = 5a-1 for each element in A.
Determine f(-1).
-6
Let A = {@,#,*} and B = {0,1,2}.
Suppose that f is the relation from A to B defined by
f = {(@,0),(#,1),(*,1)}. Is f a function from A to B?
Yes
Consider the floor function f(a) = └ a ┘.
Then f(-0.1) = 0.
False
Find a formula for the nth term of the sequence 5,10,15,20,25,... .
a subscript n = 5n
A correct way to express the sum of the numbers in the finite sequence 3, 4, 5, 6, 7 would be as
7
∑ i
i=3
True
If we encrypt the word TROY with a shift cipher with k = 3, what would the corresponding letters of the ciphtertext be? (use capital letters)
WURB
The 3rd and 4th terms of the Fibonacci sequence are 5 and 8, respectively.
False
Let A = {1,2}, B = {g,r,a,y}, and f:A -->B is the function defined by f = {(1,g),(2,g)}. Determine the domain, codomain, and range of f.
The domain is A, codomain is B, and the range is {g}
Consider the sequence whose formula for the nth term is given by the formula
a subscript n = n / 5.
Then the term a subscript 4 = 1.
False
Consider the function f:Z→B, where B = {0,1,2,3}, defined by f(a) = a mod 4. Compute f(25) and f(-1).
f(25) = 1 and f(-1) = 3.
Consider a sequence (as typically defined) defined by a sub n = 2a sub n-1, for n = 2, 3, 4, ... ,
and where a sub 1 = -2.
Determine a sub 4.
-16
In using the division algorithm to compute the quotient and remainder for a = 25 and b = 9, the quotient is 2 and the remainder is 7.
True
Compute the sum
5
∑ (2i-1)
i=1
25
The double sum
2 3
∑ ∑ (i*j) = 18
i=1 j=1
True
In performing a trace of the Recursive Euclidean Algorithm to compute the gcd of a = 12 and d = 8 (using the naming conventions presented in the pseudocode), which of the following best represents what the output of the algorithm would be?
4
Suppose we were to perform a trace of the algorithm presented in the textbook to convert 110002 to base 10. What is the value of the variable "sum" after passing through the i = 2 stage of the "for" loop?
0
Suppose that, in performing a trace of the Recursive Euclidean Algorithm on 20 and 12, if we set a = 20 and d = 12, per the naming conventions in the pseudocode. When we first go into the "If" condition where it states "If a mod d = 0," then the statement "20 mod 12 = 0" has what truth value?
False
Which of the following best represents a trace of the pseudocode recursive factorial algorithm with n = 2?
2! = 2 x (RFA with n = 1)
= 2 x (1 x (RFA with n = 0))
= 2 x (1 x 1))
= 2 x 1
= 2
The base 16 integer CAB16 converted to base 10 is 4423.
False
In tracing the Recursive Fibonacci Sequence Algorithm, the value of the term F6 would work out to be 13.
False
Convert 1111101 to base 16.
7D
Suppose we were to perform a trace of the algorithm presented in the textbook to convert 11000sub2 to base 10. Then, we would name the input a4=1, a3=1, a2=0, a1=0, and a0=0; also n = 4.
True
Convert 15010 to base 2.
10010110
The output to the recursive factorial algorithm with n = 4 is 24.
True
Consider the following partial grade book for a certain class.
Name | Exam 1 Score
-------------------------
Bill | 70
Gene | 83
Jesse | 86
-------------------------
Consider the set A = {Bill, Gene, Jesse} and B = {0,1,... , 100}. Then consider the relation R from A to B consisting of pairs (a,b) from the table above with a ∈ A, b ∈ B. In listing all the pairs of R, we have {(70,Bill),(83,Gene),(86,Jesse)}.
False
Let A = {1,2,3} and B = {0,1,2,3}. If R is the ">" relation from A to B, list the pairs in R.
{(1,0),(2,0),(2,1),(3,0),(3,1),(3,2)}
Let A = {0,2,4} and B = {0,1,2}. Let R be the relation from A to B of all ordered pairs of the form (a,b), where a is in A and b is in B, and where a/b is an integer. Then the pairs in R are {(0,1),(0,2),(2,1)(2,2),(4,1),(4,2)}
True
Let A = {0,2,4} and let R be the relation from A to A consisting of pairs (a,b), where a ≤ b. List all the pairs in R.
{(0,0),(0,2),(0,4),(2,2),(2,4),(4,4)}
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,0),(1,2),(2,1),(2,2)}. Is R reflexive?
No
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}. Is R transitive?
No
Let A = {1,2,3}, B = {2,3}, and C = {3,6}. Suppose R is a 3-ary relation that is defined by the following rule (a,b,c) is an element of R provided that a + 2 = b and b | c, and where a ∈ A, b ∈ B, and c ∈ C. Then the triples in R are {(1,3,3),(1,3,6)}.
True
Let A = {0,3,6,9}, and let R be the relation on A such that(a,b) ∈ R provided a | b, where a and b are both elements of A. List all pairs in R.
{(3,3),(3,6),(3,9),(6,6),(9,9),(3,0),(6,0),(9,0)}
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,2),(2,2)}. Is R antisymmetric?
Yes
Let A = {3,4,5}, and let R be the relation on A consisting of pairs of the form (a,b), where ab > 15, and both a and b are elements of A. If we represent R using a matrix (where the first, second, and third rows/columns of the matrix correspond to 3, 4, and 5, respectively), the matrix would be
[ 0 0 0 ]
0 1 1
[0 1 1 ]
True
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(1,0),(2,2)}. Is R symmetric?
Yes
Consider the following partial table of employee names, employee identification (ID) number, department, and age.
What would be the primary key?
Employee ID
Let A = {1,2,7} and consider the relation R on the set A defined by R = {(1,2), (1,7), (7,1)}. In representing R using a digraph, there should be a directed edge from 2 to 1.
False
Consider the matrix
[ 1 1 0 ]
1 0 1
[ 0 0 0 ]
representing a relation R on the set {x,y,z}. List all the pairs in this relation, where row/column 1 corresponds to x, row/column 2 corresponds to y, and row/column 3 corresponds to z.
{(x,x),(x,y),(y,x),(y,z)}
Let A = {0,1,2} and B = {4,8,9}. Let R be the relation from A to B such that (a,b) ∈ R provided that ab = 8. Then all the pairs in R are {(8,1),(4,2)}
False
Let P be the statement "6 is even" and Q be the statement "1 ∈ {2,3,4}".
Determine the truth value of P ^ Q
False
Let P(x) be the open sentence "x^3 = 4," where the domain for x is the set S = {2,3}. What is the truth value of ∃xP(x)?
False
Suppose a variable b has been assigned to the value of 5, and a loop has the form "While (b > 4) Do ... ."
Will we be granted access into this loop?
Yes
The statement ¬P v ¬Q is logically equivalent to the statement ¬(P v Q).
False
Let P(x) be the open sentence "x has three sides," where the domain for x is the set of all triangles. Provide a translation of the statement ∀xP(x).
"Every triangle has three sides."
Let P be the statement "Triangles have five sides and Q be the statement "1 < 2." Determine the truth value of P ⊕ Q
True
Let P(x) be the open sentence "x + 1 < 3" where the domain for x is the set S = {-1,0,1}. What is the truth value of ∀xP(x)?
True
Is "2 >7" a statement?
Yes
Let P be the statement "The set {a,b,c} has exactly 4 elements" and Q be the statement "3 ∈ {1,2,4}". Determine the truth value of the statement P ⇒ Q.
True
Let P be the statement "The sun is made of lemons" and Q be the statement "8 > 1." Express the statement P v Q in words.
"The sun is made of lemons or 8 > 1."
Let P be the statement "The moon is made of cheese" and Q be the statement "19 ∈ {1,2,3,4,5,6,7,8,9,10}". Determine the truth value of P ⇔ Q.
True
Is "3x = 9" a statement?
no
Which (if any) entries in the last column (the ones in boldface) of the table below are incorrect?
P | Q | P⇒Q | (P⇒Q)⇒P
--------------------------
T | T | T |
T
T | F | F |
T
F | T | T |
T
F | F | T |
T
The entries in boldface in the last two rows.
Let P be the statement "A square has seven sides" and Q be the statement "A triangle has one side." Write the statement P ⇒ Q in words.
"If a square has seven sides, then a triangle has one side."
Let P be the statement "9 ÷ 0 is undefined". Express the statement ¬P in words and determine its truth value.
"9 ÷ 0 is defined." False.
Let A = {3,4,5} and B = {g,o}. Then A x B = {(3,g),(3,o),(4,g),(5,g),(5,o)}.
True
Suppose we are given the following list of Prolog facts.
player(Smith, TROY)
player(Jones, TROY)
player(Ware, TROY)
player(Roblee, ACME College)
Then the query ?player(X, ACME College) would return Yes
False
Let P(x) be the statement "x^2 < 0," where the domain for x is the set S = {1,2,3}. What is the truth value of ∃xP(x)?
False
Let A = {0,1} and B = {1,2}. If R is the "<" relation from A to B, list all the pairs in R.
{(0,1),(0,2),(1,2)}
Let A = {1,2,3}, B = {3,4}, and C = {5,6}. Suppose that R is a 3-ary relation that is defined by the following rule: (a,b,c) is an element of R provided that a + 1 = b, and b + 2 = c, where a ∈ A, b ∈ B, and c ∈ C.Then the triples in R are {(2,3,5),(3,4,6)}.
True
Which (if any) entries (in boldface) in the last column of the table below are incorrect?
--------------------------------
P | Q | ¬P | ¬P v Q
--------------------------------
T | T | F |
T
T | F | F |
T
F | T | T |
T
F | F | T |
T
The entry in boldface in the second row.
Find the bitwise OR of the following two bit strings: 0111, 1010.
1111
Let P be the statement "1 + 1 = 5" and Q be the statement "1 < 2." Express the statement P ^ Q in words.
"1 + 1 = 5 and 1 < 2."
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,1),(1,0),(1,2),(2,0),(2,1)}. Is R reflexive?
No
Let A = {4,{4}} and B = {4,{4},{4,{4}}}. Is A ⊆ B?
Yes
Let P be the statement "A triangle has four sides" and Q be the statement "A triangle has three sides." Determine the truth value of P ⇒ Q.
True
Let A = {2,3,4} and R be the relation on A given by R = {(2,2),(2,3),(3,2),(3,4),(4,2)}. Is R transitive?
No
Consider the following partial table of customer names, account number, and age. Then account number is a primary key.
True
Consider the relation R = {(1,1),(1,3),(2,1),(2,2),(2,3),(3,1)} on the set A = {1,2,3}. In using a digraph to represent R, then there should not be a directed loop from 1 to itself.
False
Let A = {g,o,l,f,e,r} and B = {g,l,f,1}. Find A - B.
{o,e,r}
Let P be the statement "5 + 6 = 11." Express the statement ¬P in words and determine its truth value.
"5 + 6 ≠ 11." The truth value is false.
Consider the matrix
[0 1]
[0 1]
representing a relation R on the set {x,y}. List all pairs in the relation, where row/column 1 corresponds to x and row/column 2 corresponds to y.
{(x,y),(y,y)}
Consider the following partial grade book for a certain class.
--------------------------
Name | Midterm Score
Maudie | 87
Truman | 81
Vera | 77
--------------------------
Consider the set A = {Maudie, Truman, Vera} and B = {0,1, ... , 100}. Then consider the relation R from A to B consisting of pairs (a,b) from the table above with a ∈ A, b ∈ B.
In listing all the pairs of R, we have {(87,Maudie), (81, Truman), (77,Vera)}.
False
Is "3 + 1 = 6" a statement?
Yes
The cardinality of the set {6, 7, {6,{6}}, {7} ,{6, 7}} is 5.
True
{4,5} ∈ {4,5,{4},{5}}
False
Let A = {a,b,c,1} and B = {1,b,c,v}. Find A∩B.
{b,c,1}
Suppose we are given the following list of Prolog facts.
player(Smith, TROY)
player(Jones, TROY)
player(Ware, TROY)
player(Roblee, ACME College)
Then the query ?player(X, TROY) would return Smith, Jones, Ware.
True
Is Ø ⊆ {e,m,p,t,y}?
Yes
Let P be the statement "The moon is made of cheese" and Q be the statement "Florda has beaches." Determine the truth value of P ⇔ Q.
False
{0, 1, 1} = {1, 1, {0}}
False
In completing the truth table for the statement (P⇒Q) v Q, in the row where P is false and Q is false, what is the value of the statement (P⇒Q) v Q?
True
Let P(x) be the open sentence "x has three sides," where the domain for x is the set of all squares. Provide a translation of the statement ∀xP(x).
"Every square has three sides."
Let U = {g,o,l,f,e,r} and A = {f,o,r,e}. Find A^c.
{g,l}
Suppose a variable x has been assigned to the value of -3, and a loop has the form "While (x > -2) Do ... ." Will we be granted access into this loop?
No
Let P be the statement "1 + 1 = 3" and Q be the statement "1 < 3." Determine the truth value of P ^ Q.
False
Let A = {0,1,2} and R be the relation on A given by R = {(0,0),(0,1),(0,2),(1,1),(1,0),(1,2),(2,0),(2,1),(2,2)}. Is R symmetric?
Yes
Let A = {0,1,2} and B = {5,6}. Let R be the relation from A to B such that (a,b) ∈ R provided that a + b = 6. Then all the pairs in R are {(0,6),(1,5)}.
True
Let A = {4,5,6,7} and B = {6,c,h,i,p}. Find A ∪ B.
{4,5,6,7,c,h,i,p}
Let P be the statement "1 = 2" and Q be the statement "1 > 2." Determine the truth value of P⇒Q.
True
A board is 4.6 feet long. How many meters long is the board?
1.4
Top Cat Car Wash buys liquid soap in 5 gallon buckets. How many liters are in one 5 gallon bucket
18.9
Greg plowed for 2.5 hours in the morning, 1 hour and 15 minutes in the afternoon, and 1 hour and 30 minutes in the evening. How many total hours did Greg plow?
NOT: 15.5
The diagram shows the dimensions, in meters, of Alice's workshop. To determine what size heater to buy for the workshop, Alice must compute the surface areas of each type of surface on the outside walls of the workshop. What is the surface area of the doors shown in the diagram?
NOT: 13
You need to buy fertilizer for a circular flower bed with a diameter of 13 feet. If one bag will fertilize 10 square feet, how many bags do you need to buy?
14
The admissions department at Springfield Technical College is reviewing applications to attend the college. They know that only about 25% of the students who are accepted will choose to attend. They would like to have 300 new students. How many students should they accept?
NOT: 75
80% of the 350 employees at Animal Kingdom Park have college degrees. How many employees at the park have college degrees?
280
baker uses 2 cups of sugar, how much flour should the baker use?
2 2/3
The Wholesome Egg Company sells eggs in three different size cartons. Which carton offers the lowest price per egg?
Cartons B and C are the lowest
A watch gains 18 seconds in a day. How many minutes will the watch gain in 3 weeks? Which expression can be used to solve this problem?
NOT: 18 x 21 x 60
Angela is plowing a large field for spring planting. On the first day, she plowed of the field. On the second day, she plowed of the field. What fraction of the field is NOT plowed at the end of the second day?
4/21
The scale factor on a scale drawing of machine part is . If the part is inches long on the drawing, how long is the actual part?
NOT: 19
A computer diagnostic program has completed of its analysis. If the analysis requires 1 second to complete of the analysis, in how many seconds will the program finish?
2 seconds
Dallas needs liter of milk for a recipe, and there is of a liter of milk left in the carton. How much is left after Dallas makes his recipe?
1/20 of a liter
The area of a rectangular bathroom floor is feet. If the bathroom is feet wide, what is the length?
8 1/4
A bottle contains ounces of medicine. The veterinarian prescribes ounce of medicine per day. How many days will the bottle last?
21 days
A floor plan shows the area of a room is 271.25 square feet. If the room is 15.5 feet wide, how long is it?
17.5
The triangular gable on this house is 8 feet tall. The area of the gable is 120 square feet. What is the length of the base?
NOT: 15
Kerry is conducting a marketing survey. She mailed 45,128 surveys, and received 17,892 responses. Approximately what percentage of people responded to the survey?
NOT: 27%
Louis found two bakeries to provide bagels for his sub shop. The first bakery offers 350 bagels for $168.00 and the second bakery offers 475 bagels for $209. How much will Louis pay for 800 bagels if he buys from the bakery with the lower price?=
$352.00
Once a week, 50 employees have pizza delivered for lunch. Pizza Villa charges a delivery fee of $20 plus $3.00 for each personal-size pizza. Slice of Heaven Pizzeria charges $2.50 for each personal-size pizza and $30 for delivery. Which caterer offers the better deal?
Slice of Heaven Pizzeria
Reggie is paid by commission and earned $1,575 last month. This month he made $1,250. By what percentage did Reggie's earnings decrease? Round your answer to the nearest whole percentage.
NOT: 20%
A hotel currently has 280 rooms. The management company decides to build a new wing that will increase the capacity to 405 rooms. By what percentage will this expand the capacity of the hotel? (Round to the nearest tenth of a percent.)
44.6%
Marissa is planning refreshments for a day-long conference. At her last event, there were 240 people and they drank 18 gallons of coffee. There are 320 people signed up for this conference, how many gallons of coffee will she need?
24 gallons
Kevin lives 19 kilometers from his office. It costs Kevin $0.31 per mile to drive his truck. Kevin made the equation below to determine how much it costs to drive his truck from his house to the office. Cost = 19 x 0.62 x 0.31 Which statement describes the accuracy of Kevin's equation?
A
A fish market advertises fresh fish for $4.62 per kilogram. The original price was $5.25 per kilogram. By what percentage did the price decrease?
12.0%
A lawn mower uses 0.7 gallons of gas every 3 hours. The gas tank holds 2 gallons. How long can the mower run on a full tank?
8.57 hours
Tim is a wildlife biologist studying deer parasites. During a study, he finds that 12 out of 32 deer were affected by parasites. How many of the 1200 deer in the local population would he expect to be affected by parasites?
450
A segmental paver is installing a circular patio that has a radius of 12 feet. How much edging material will be needed for the circumference of the patio (to the nearest tenth)? (Use 3.14 for Pi.)
NOT: 37.68 feet
A sonographer is trying to determine the size of a gallstone. If the stone is spherical and has a diameter of 12 millimeters, what is its volume?
0.9 cubic centimeters
At the local animal shelter, there is a cone-shaped dog-food dispenser. If the dispenser is 54 centimeters tall and has a diameter of 24 centimeters at the top, what is the volume (to the nearest cubic centimeter)?
8,139 cubic centimeters
The marketing department of a cereal company proposes to reduce the size of a container of oatmeal to avoid raising the price. The package is currently a cylinder that is 8 inches tall and has a diameter of 4 inches. If both dimensions are reduced by one-half inch, by what percentage is the volume of the container reduced? (Round your answer to the nearest percentage.)
NOT: 72%
The CoffeeShop is creating a new blend of coffee. Coffee A sells for $5 per pound. Coffee B sells for $9 per pound. The CoffeeShop will mix 20 pounds of the new blend and sell it for $6 per pound. The following equations represent the problem of keeping the cost of the blend consistent with the ingredients. a + b = 20 5a + 9b = 120 a = pounds of Coffee A and b = pounds of Coffee B How many pounds of Coffee B should be used in the blend?
NOT: 12 or 8
A recipe requires cup of milk. To make the recipe for a large crowd, Marcos needs cups of milk. One gallon of milk costs $2.49. Marcos made the equation below to find how much it will cost to buy milk for the recipe. Cost = ( ÷ 4) x 2.49 Which statement describes the accuracy of Marcos' equation?
Marcos should divide the answer by 4
Joni is placing an ad in a local paper. Her original design for the ad is 6 inches wide and inches tall, but the space available is only inch wide. If she shrinks the design to fit, how tall will the ad be?
NOT: 5 1/2
The diameter of a ball is 6 inches. What is the volume of the ball?
113.04 in^3
individuals
entities described by a set of data. can be people, groups, animals, or things
variable
particular characteristic or trait that can take on different values for different individuals
qualitative
categorical value
quantitative
numerical value
distribution
gives information about how often the variable takes a certain value or intervals of values
frequency distribution
states all observed values of the variable and how many times the variables takes on each of these traits
relative frequency distribution
states all observed values of the variables and what fraction of the time they occur
grouped frequency distribution
gives how many times in each interval of values the variable takes on a particular value
histogram
a graph of the distribution of outcomes for a single numerical value
outlier
individual that falls outside the overall pattern
right-skewed distribution
longer tail of histogram on right
left-skewed distribution
longer tail of histogram on left
symmetric
left and right side mirror each other
mean
average
median
middle value
mode
most frequently occurring value
range
measure of variability
quartile
describes which values fall some quarter of the way through the observations
five number summary
minimum, Q1, median, Q3, maximum
boxplot
graph of the five number summary
standard deviation
average amount of observations are distant from the mean
normal curve
a curve such that the area under the curve is one.
normal distribution
described by a normal curve
response variable
measures an outcome or result of a study (y)
explanatory variable
a variable that we think explains or causes a response variable (x)
scatterplot
graph of plotted points.
how to describe scatterplot
form, direction, strength
positive association
if changes tend to be in same direction they both increase or decrease
negative association
if changes tend to be in opposite direction. one increases as other decreases
regression line
straight line that describes how a response variable y changes as an explanatory variable x changes. a regression line is used to predict the value of y for a given x.
correlation
measures the direction and strength of the straight-line relationship between two numerical values
least-squares regression line
line that makes the sum of the quares of the vertical distances of the data points from the line the least value possible.
regression line slope is
m= r * sy/sx
population
entire group of individuals about which we want information
sample
is a part of the population from which we actually collect information
convenience sample
sample of individuals who are selected because they are members of the population who are easy (convenient) to reach
bias
systematically favoring certain responses
voluntary response sample
consists of people who choose themselves by responding to a general appeal
simple random sample (SRS)
of size n consists of n individuals from the population chosen in such a way that ever set of n individuals has an equal chance to be chosen
undercoverage
occurs when some groups in the population are left out of the process of choosing a sample
nonresponse
occurs when an individual chosen for the sample can't be contacted or refuses to participate
experiment
deliberately imposes a treatment on individuals to observe their response
confounded
when effects cannot be distinguished from each other
statistically significant
observed effect that is so large it would rarely occur by change (less than 5% of the time)
placebo effect
people often show a response because they perceive they should
double-blind
an experiment where the subject or the people interacting with them know whether the subject is receiving treatment or a placebo effect
observational study
does not try to manipulate the environment; it simply observes the measurements of variables of interest that result from people's free choices
prospective study
finds a group of subjects, both who receive and don't receive treatment, and follows them forward in time
retrospective study
finds a group of subjects who have already had treatment and response and looks backward in time for the information
statistical inference
refers to methods for drawing conclusion about an entire population on the basis of data from a sample
parameter
is a fixed (usually unknown) number that describes a population
statistic
a number that describes a sample
sampling distribution
of a statistic is the distribution of values taken on by the statistic in all possible samples of the same size from the same population
confidence interval
the interval within which we are __% sure that our true proportion lies
margin of error
is half the width of the confidence interval
product rule
if a procedure can be broken down into a sequence of 2 tasks then the number of ways to do that task is n1 * n2
sum rule
if a takes can be either done in 1 of n1 ways or 1 of n2 ways then there are n1 + n2 ways of doing that task
subtraction rule
if a task can be done in either n1 ways or n2 ways then the number of ways to do the task is n1 + n2 minus the number of ways to do the task in both ways
division rule
seating people around a circle, there are n!/n ways to do it
pigeonhole principle
if k is a positive integer and k + 1 or more objects are placed into k boxes then there is at least one box containing 2 or more of the objects
generalized pigeonhole principle
if N objects are placed into k boxes then there is at least one box containing ceiling(N/k) objects
permutation
an ordered arrangement of the elements in a set
r-permutation
an ordered arrangement of r elements in a set
P(n,r)
the number of r-permutations of a set with r elements
C(n,r)
the number of r-combinations of a set with n elements
Pascal's identity
(n+1/k)=(n/k-1)+(n/k)
Declarative Sentence
A sentence that declares a fact
Proposition
A declarative sentence that is either true or false, but not both.
EX:
1 + 1 = 2.
Toronto is the capital of Canada.
NOT an example:
What time is it?
Read this carefully.
x + 1 = 2.
Propositional variables
Variables that represent propositions, just as letters are used to denote numerical variables.
Conventional letters used are:
p, q, r, s ...
Truth value
Denoted by T if it is a true proposition
Denoted by F if it is a false proposition
Propositional Calculus
or
Propositional Logic
Area of logic that deals with propositions
First developed by Aristotle
Compound Propositions
New propositions that are formed from existing propositions using logical operators
Logical operators
and, or, not
words which combine or modify simple propositions to make compound propositions.
Negation of p
¬p (or p with a bar/line on top)
TRUTH VALUE: true if and only if p is false
The truth value of ¬p is the opposite of the truth value of p.
ENGLISH EXPRESSION: "not p" or "it is not the case that p"
Example:
Proposition: "Michael's PC runs Linux"
Negation: "It is not the case that Michael's PC runs Linux"
(adds standard 'Is not the case that')
Proposition: "David's phone has at least 32GB of memory"
Ex 1: "It is not the case that David's phone has at least 32GB of memory."
(adds standard 'Is not the case that')
Ex 2: "David's phone does not have at least 32GB of memory" - (adds 'does not' & replaces 'has' with 'have')
Ex 3: "David's phone has less than 32GB of memory"
(replaces 'at least' with 'less than')
Truth Table
A table used as a convenient method for organizing the truth values of statements
A list of all possible input values to a digital circuit, listed in ascending binary order, and the output response for each input combination.
Negation Operator
This constructs a new proposition from a single existing proposition.
If I'm reading this horrible book correctly, it's basically the ¬ in front of the proposition variable.
Connectives
Logical operators that are used to form new propositions from two or more existing propositions.
"and" "or" "if" & "if and only if"
Conjunction
And connective. "p and q"
TRUTH VALUE: true if and only if both p and q are true
ENGLISH EXPRESSION: "p and q" "p but q"
p∧q: p is true and q is true
The conjunction p ∧ q is true when both p and q are true and is false otherwise.
p: Dave's dream car has more than 3 doors
q: Dave's dream car has more than 5 cylinders
p∧q : p and q : Dave's dream car has more than 3 doors and Dave's dream car has more than 5 cylinders.
Ex #1: Dave's car has 4 doors and has 6 cylinders. then p and q is true (when both p and q are true)
Ex #2: Dave's car has 4 doors and 4 cylinders. The first part is true to proposition p but false to proposition b, so the conjunction of both of p and q is false (because one of them is false).
It's like if someone at work says he'll be only happy if the car is BOTH at least 4 door and 6 cylinder. If you find a car that is 4 door but only 4 cylinder, you haven't met his condition.
Disjunction
(Inclusive) Or connective. "p or q"
TRUTH VALUE: false if and only if both p and q are false.
ENGLISH EXPRESSION: "p or q (or both)" "p unless q"
p∨q: p is true or q is true, or possibly both
The disjunction p ∨ q is false when both p and q are false and is true otherwise.
p: Dave's dream car has more than 3 doors
q: Dave's dream car has more than 5 cylinders
p∨q: Dave's dream car has more than 3 doors or Dave's dream car has more than 5 cylinders.
Ex. Dave's car has 4 doors and has 4 cylinders. p is true while q is false. However, this proposition is p or q, so only one needs to be true (satisfied) for it to be true.
It's like if someone at work says he'd be happy if the car has 4 doors or is a 6 cylinder. You found a car that meets one of those conditions so you've satisfied p∨q, p or q.
Conditional or Implication
If connective
TRUTH VALUE: false if and only if p is true and q is false.
ENGLISH EXPRESSION: (see bottom)
p→q: If p is true, then q is true
p → q is false when p is true and q is false, and true otherwise.
In the p → q, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).
p: I am elected
p: I will lower taxes
p → q = Plainspeak: If I am elected, then I will lower taxes.
Truth Table Example:
He is elected | He lowers taxes = True
He is elected | He doesn't lower taxes = False
He is not elected | He lowers taxes = True
He is not elected | He doesn't lower taxes = True
If he is elected, then you expect him to lower taxes. If elected and does not lower taxes, that is a failure (false). If he is not elected, it is not expected so those values are true whether he does or does not.
Easier Truth Table Example:
p: 100% on final
q: A grade
p→q results
100% on final | A grade = True (result)
100% on final | not A grade = False (result)
not 100% on final | A grade = True (result)
not 100% on final | not A grade = True (result)
ENGLISH EXPRESSION:
"if p, then q"
"p implies q"
"p only if q"
"p is sufficient for q"
"a sufficient condition for q is p"
"q if p"
"q whenever p"
"q when p"
"q is necessary for p"
"a necessary condition for p is q"
"q follows from p"
"q unless ¬p"
"q unless not p"
"q or else not p"
Exclusive Or
p ⊕ q
TRUTH VALUE: exactly one of p and q is true.
ENGLISH EXPRESSION: "either p or q" "p or q but not both"
The proposition that is true when exactly one of p and q is true and is false otherwise.
p: Dave loves babies
q: Dave hates babies
It would not make sense if Dave both loved and hated babies, so obviously when both p & q are true, the result is false.
p: Students who have taken calculus
q: Students who have taken computer science
A school may want to run an 'exclusive or' on their students to see who would be eligible for taking this class that only allows someone to have taken one or the other, but not both.
Example of that truth table:
Taken calculus & taken computer science = false
Taken calculus & not taken computer science = true
Not taken calculus and taken computer science = true
Not taken calculus and not taken computer science = false
Biconditional
If and only If connective
TRUTH VALUE: true if and only if p and q have same truth value
ENGLISH EXPRESSION:
"p if and only if q" or "p iif q"
"p is necessary and sufficient for q"
"if p, then q, and conversely"
p↔q: p is true if and only if q is true, or p is equivalent to q
NOTE: The statement p ↔ q is true when both
the conditional statements p → q and q → p are true and is false otherwise. That is why we use
the words "if and only if" to express this logical connective and why it is symbolically written
by combining the symbols → and ←.
This is essentially an if, then statement.
"If you finish your meal, then you can have dessert."
"You can have dessert if and only if you finish your meal."
ENGLISH TRUTH TABLE:
q: finish your meal
p: can have desert
If you finish your meal, you can have your desert = TRUE
If you finish your meal, you can NOT have your desert = FALSE
If you do NOT finish your meal, you can have your desert = FALSE
If you do NOT finish your meal, you can NOT have your desert = TRUE
Converse
Given: p → q
This would be: q → p
Example:
We will use the conditional verbiage "q when p"
p: Home team wins
q: It is raining
p → q = Home team wins "when" it is raining
q → p = It is raining "when" the home team wins
Contrapositive
Given: Given: p → q
This would be: ¬q → ¬p
This always has the same truth table as p → q
Example:
We will use the conditional verbiage "q when p"
p: Home team wins
q: It is raining
¬q → ¬p = It is not raining "when" the home team does not win.
Inverse
Given: p → q
This would be: ¬p → ¬q
Example:
We will use the conditional verbiage "q when p"
p: Home team wins
q: It is raining
¬p → ¬q = Home team does not win "when" it is not raining.
Equivalent
When two compound propositions always have the same truth value.
Converse & Inverse are this
The conditional and contrapositive are this.
p → q ≡ ¬q → ¬p
q → p ≡ ¬p → ¬q
p → q ̸≡ q → p (should be the symbol for not equal)
p ⊕ q ≡ ¬(p ↔ q)
Logical connectives (operators) table
Precedence
1) ¬
2) ∧
3) ∨
4) →
5) ↔
tautology
A compound proposition that is always true, no matter what the truth values of the propositional
variables that occur in it
contradiction.
A compound proposition that is always
false
contingency.
A compound proposition that is neither a tautology nor a
contradiction
logically
equivalent.
Compound propositions that have the same truth values in all possible cases
¬(p ∧ q) ≡
¬p ∨¬q
¬(p ∨ q) ≡
¬p ∧¬q
Distributive Law of disjunction over conjunction
p ∨ (q ∧ r)≡
(p ∨ q) ∧ (p ∨ r)
Identity laws
p ∧ T ≡ p
p ∨ F ≡ p
Domination laws
p ∨ T ≡ T
p ∧ F ≡ F
Idempotent laws
p ∨ p ≡ p
p ∧ p ≡ p
Double negation law
¬(¬p) ≡ p
Commutative laws
p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p
Associative laws
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Distributive laws
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
De Morgan's laws
¬(p ∧ q) ≡ ¬p ∨¬q
¬(p ∨ q) ≡ ¬p ∧¬q
Absorption laws
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Negation laws
p ∨¬p ≡ T
p ∧¬p ≡ F
Logical
Equivalences Involving
Biconditional Statements.
p ↔ q ≡ (p → q) ∧ (q → p)
p ↔ q ≡ ¬p ↔¬q
p ↔ q ≡ (p ∧ q) ∨ (¬p ∧¬q)
¬(p ↔ q) ≡ p ↔¬q
Boolean Logic
0 and 1, false and true, no and yes, low and high
Operations of Boolean Values
NOT (Negation), AND, XOR
NOT (Negation)
Opposite of original value
OR
Must meet one requirement to be true
AND
Must meet both requirements to be true
Conditional Operations
If, then or >=
Exclusive Or Operation (XOR)
If both are true, answer is false
Equivalence Operation
...
Precedence Order
Order of execution matters
Order of Boolean Operations
NOT -> AND -> XOR -> OR -> CONDITIONAL -> EQUIVALENCE
Half-Adder
0 + 0 = 0, 0 + 1 or 1 + 0 = 1, 1 + 1 = 2 -> (10)base2 -> A
Closure
If P and Q are boolean expressions, then NOT P, P AND Q, P OR Q
Associative
Same as associative property in chapter 1
Commutative
Same as commutative property in chapter 1
Distributive
OR distributes over AND, AND distributes over OR
Identity
P OR FALSE = P, P AND TRUE = P
Inverses
Every boolean expression P has a negation NOT P such that, P OR NOT P = TRUE, P AND NOT P = FALSE
Less Than (<)
If A < B, A < B -> TRUE
Equal
If A = B, A = B -> TRUE
Greater Than or Equal
IF A >= B, A >= B -> TRUE
DeMorgan's Law
If A and B is NOT TRUE, then at least 1 is FALSE
Ten's Complement
4-digit machine. A number X and it's negative -X sum to 0
Small Numbers (Ten's)
0 - 4,999 have their normal values
Large Numbers (Ten's)
5,000 - 9,999 are all negative
p → q
!p ∨ q
!q → !p
p ∨ q
!p → q
p ∧ q
!( p → !q)
!(p → q)
p ∧ !q
(p → q) ∧ (p → r)
p → (q ∧ r)
(p → r) ∧ (q → r)
(p ∨ q) → r
(p → q) ∨ (p → r)
p → (q ∨ r)
(p → r) ∨ (q → r)
(p ∧ q) → r
p ↔ q
(p → q) ∧ (q → p)
!p ↔ !q
(p ∧ q) ∨ (!p ∧ !q)
!(p ↔ q)
p ↔ !q
!∃xP(x)
∀x!P(x)
!∀xP(x)
∃x!P(x)
roster method
set = {a, b, c, d}
N
Natural Numbers {0,1,2,3...}
Z
Integers
Z+
Positive Integers (not including 0)
Q
Rational Numbers {p/q | p and q domains Z}
R
Real Numbers
R+
Positive Real Number (not including 0)
C
Complex Numbers
intervals
[a,b] [a,b) (a,b] (a,b)
closed interval
[a,b]
open interval
(a,b)
empty/null set
denoted by Ø, nothing in it { }
singleton set
has one element (sets within sets can be thought of as folders on computer)
paradoxes
logical inconsistencies
universal set (U)
all other objects (represented by rectangle in venn diagram)
proper subset(or superset)
A ⊂ B if and only if there's an element x of B that isn't an element of A (otherwise A ⊆ B)
prove two sets to be equal
show A ⊆ B and B ⊆ A
finite set
S is a set. Exactly n distinct elements. n is Z+0.
cardinality of S (|S|)
number of distinct elements, n
set is infinite
set is not finite
every set is guaranteed:
(1) ∅ ⊆ S (2) S ⊆ S
subset
A is a subset of B if and only if every element of A is an element of B (A ⊆ B)
superset
A is a superset of B if and only if every element of B exists in A (A ⊇ B)
ordered n-tuple
(a1, a2,...,an) is ordered collection with a1 as first element and so on...
cartesian product (A x B)
set of all ordered pairs (a,b)
A×B ={(a,b) | a ∈ A ∧ b ∈ B}
truth set
predicate P, domain D, set of elemenet x in D for which P(x) is true {x ∈ D | P(x)}
What is the sample space S
The sample space S of a random experiment is the set of all possible outcomes of the experiment.
Show me the sample space S of a die-tossing experiment?
S = { 1,2,3,4,5,6}
Show me the sample space S of a coin-flipping experiment?
S = { H, T }
What is an event?
A subset of a sample space is called an event.
Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
Explain a Simple event?
Simple events are the 1-element subset of the sample space. Simple events are denoted by capital letters.
Union
A∪B of two events A and B whenever A occurs, or B occurs.
Intersection
A∩B of two events A and B occurs whenever A and B occurs.
Complement
A' of an event A occurs whenever the event A does not occur.
Finite probability space
The probability of an event iss an estimate of the frequency of occurrence of that event when the experiment is repeated a large number of times.
Equiprobable space
mutually exclusive events
The Additional Law of Probability
Properties of Probabilities
Given events A and B where P(B)>0
Independent events
Independent events A and B
The multiplication law of probability
Probability of the Union of two events
Conditional Probability
Compound Experiment
Repeated Trials ( Bernoulli Trials)
Binomial Probability
Base's Formula (Supplement)
Combination Formula
Random Variables and Expected Value
Expected Value (Mean)
Expected Mean example
Median
Median and Mode
Variance
Binomial Distribution
The Principle of Mathematical Induction
Sigma
Summation Notation Properties
Binomial Theorem
Distribution of a random Variable
Expected Value E(X)
Standard Form Deviation of Variance
Example of how to find the variance and Standard of a deviation
Standard Deviation of a random experiment
path on a graph
a sequence of edges and vertices which are all connected
valence
number of edges attached to a vertex
Hamiltonian Circuit
a path on a graph which starts and stops at the same vertex and does not cross each vertex more than once
digraph
directional graph
feasible region
collection of all possible solution with an area/region
weighted graph
a graph which has weights/numbers on each of its edges
bar graph
a graph which uses bar lengths which indicate the percentages of each subject in their defined categories
median
middle number of a set after arranging it from smallest to largest
mode
number which appears most often in a given set of numbers
sample space
the set of all possible outcomes in a probability problem
probability
the numerical expression of the likelihood an event will occur using the number 0-1 inclusive
random sample
the selection of a subject so that each subject has an equal chance of being chosen for a study
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