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Discrete Mathematics - Test 1
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Terms in this set (122)
Rational number
A real number, r, is called rational if it can be expressed as a ratio of two integers: r = p/q, where p and q are integers and q = 0.
Countably infinite
A set is called countably infinite if its elements can be placed in one-to-one correspondence with the positive integers. That is, every element in the set can be labeled by exactly one positive integer and every positive integer is the label for exactly one element of the set.
Discrete Mathematics
Discrete mathematics is a collection of mathematical topics that examine and use finite or countably infinite mathematical objects.
Stable assignment
Given a collection of n men, m1.....mn, and n women, w1.....wn, we wish to associate every person with a mate. Suppose that each person ranks the people of the opposite gender by preference with no ties. An assignment is one of the possible collections of n couples. An assignment is stable if there does not exist a man, mi, and a woman, wj, who are not partners but mi prefers wj to his assigned bride and wj prefers mi to her assigned groom.
Unattached, Viable
A suitor will be called unattached if he or she is not currently waiting for a suit to respond to a proposal. A suit is viable for a suitor if that suit has not already rejected a proposal from him.
Theorem 1.6: Termination
The Deferred Acceptance Algorithm always terminates.
Theorem 1.7: Stability
The Deferred Acceptance Algorithm always produces a stable assignment.
The Deferred Acceptance Algorithm
round = 1
All suitors are initially considered unattached.
while at least one suit has no pending proposal, repeat the following:
Each unattached suitor proposes to his or her string of suitors, rejecting all except the highest ranked suitor. That suitor is told to wait while the proposal is considered. (The waiting suitor therefore becomes attached for the next round.)
Add 1 to round.
end while
All suits accept the proposal of the single suitor waiting for an answer.
Theorem 1.8
The set of rational numbers is countably infinite.
Set; Element; Member; Universal Set
Informally, a set is a collection of distinct objects, each thought of as a single entity. The set is the aggregate collection. The objects in the set are called the elements or members of the set. The potential elements are considered to be members of a set U, called the universal set.
The notation x A is used to indicate that x is an element of the set A. The notation y A indicates that y is not an element of A.
Subset
A set, b, is a subset of a set, A, if every element of B is also an element of A. This is denoted B A.
Proper Subset
A set, B, is a proper subset of the set A, denoted B A, if every element of B is an element of A, but there is at least one element of A that is not an element of B.
An alternative notation, B A, is also used to emphasize the "at least one element of A that is not an element of B" part of the definition. In this book, the two notations are considered to mean the same thing.
The Empty Set,
The empty set, denoted by , is the unique set that does not contain any elements.
Set Equality
Sets A and B are said to be equal, denoted A = B, if and only if A B and B A.
Cardinality; Infinite Set
The cardinality of a set is the number of elements in the set. The cardinality of the set, S, is denoted by S .
A set, S, is said to be infinite if for each positive integer, n, there is a proper subset, Sn S, with Sn = n.
Complement
The complement, A, of the set A is the set of all elements of the universal set, x A if and only if x A.
Intersection
The intersection of the sets A and B, denoted A B, is the set of all elements that are in both A and B. That is, x A B if and only if x A and x B.
Union
The union of the sets A and B, denoted A U B, is the set of all elements that are either in A or in B (or in both). That is, x A U B if and only if either x A or x B.
Difference
The set difference, A - B, is the set of all elements that are in A but are not in B. That is, x A - B if and only if x A and x B.
Disjoint
The sets A and B are disjoint if they have no elements in common. That is, A and B are disjoint if A B = .
Union and Intersection of Multiple Sets
Let {Si | i Y} be a collection of sets with index set Y. Their union is the set
Their intersection is the set
Partition
A collection, {Si | i Y}, of nonempty subsets of a set S, is said to be a partition of S if
Cartesian Product
Let {Si | i = 1, 2, ...., n} be a fine collection of two or more sets. The Cartesian product of the collection, denoted S1 X S2 X S3 X ... X Sn, is the set of all ordered n-tuples with coordinate ai a member of Si. That is,
The Cartesian products S X S X ... X S is often abbreviated as S^n.
Power Set
Let S be a set. The power set of S, denoted P(S), is the set of all subsets of S (including the empty set and S itself).
Symmetric Difference
The symmetric difference of the sets A and B is denoted A B and is defined by
A B = (A - B) U (B - A).
Proposition 2.17
Let A and B be sets. Then A B A.
Proposition 2.18
Let A and B be sets. Then
(A - B) U (B - A) = (A U B) - (A B)
Proposition 2.19: A De Morgan Law for Sets
Let A and B be sets. Then
A U B = A B
Proposition 2.20: Eliminating Set Difference
Let A and B be sets. Then A - B = A B.
Proposition 2.21: Counting a Partition
Let S be a finite set and let {S1, S2,....,Sk} be a partition of S. Then
S = Sk
Statement
A statement is an assertion that may be labeled either true or false.
Logic operators
Objects that build compound statements from simpler statements
Tautology, Contradiction, Conditional
A statement is called a tautology if every entry in its truth table is T. A statement is called a contradiction if every entry in its truth table is F. A statement that is neither a tautology nor a contradiction is called a conditional statement.
Logical Equivalence
Two statements, A and B, are called logically equivalent if and only if A B is a tautology. Logical equivalence is denoted by the meta-operator . The definition of logical equivalence can also be stated as
A B if and only if A and B have the same truth table.
Inference; Rule of Inference
Let A and B be two statements. The B may be inferred from A, denoted by A B., if A B is a tautology. The symbol, , is the inference meta-operator. The meta-statement A B is called a rule of inference.
The Substitution Principles: Substituting an Equivalent Statement
If A B, and A is a component of a statement, C, then B may be substituted for A without changing the T/F value of C.
The Substitution Principles: Replacing a Logic Variable in a Tautology
If B is a logic variable in a tautology, C, and A is any statement, then A may be substituted for every occurrence of B in C and C will still be a tautology.
The Substitution Principles: Using a Rule of Inference
If A B, A evaluates to T, and A is a component of a statement, C, then B may be substituted for A without changing the T/F value of C.
Boolean Algebras
A Boolean Algebra, , consists of an associated set, B, together with three operators and four axioms. the binary operators + and map elements of B X B to elements of B. The unary operator complement, , maps elements of B to elements of B.
Parentheses have the highest precedence. The complement operator has the second highest precedence, followed by and then by +, the operator with the lowest precedence.
Axioms
Properties that any example of a Boolean algebra must satisfy.
Boolean Algebras: Identity Axiom
There exist distinct elements, 0 and 1, in B such that for every x B
Boolean Algebras: Complement Axiom
For every x B, there exists a unique element x B such that
Boolean Algebras: Commutativity Axiom
For every pair of (not necessarily distinct) elements x, y B
Boolean Algebras: Distributivity Axiom
For every three elements x, y, and z B (not necessarily distinct)
Boolean Expression
Let be a Boolean algebra with associated set, B. A Boolean expression over is any algebraic expression that is composed using elements from B, the operators +, , and , and variables whose possible values are elements of B.
Zero Divisor
Two nonzero elements, x and y, in an algebraic system are called zero divisors if xy = 0.
Corollary 2.31
Let x and y be elements of a Boolean algebra. Then
x + y = x iff x + y = 1.
Symmetric Difference (Boolean Algebras)
Let x and y be elements of a Boolean algebra. Then their symmetric difference is denoted x y and is defined by
Proposition 2.33
Let x and y be elements of a Boolean algebra. Then
x = y iff
Corollary 2.34
Let x and y be elements of a Boolean algebra. Then
x = y iff
Negating a Quantified Statement
To negate a universally or existentially quantified statement, interchange the quantifier and negate the predicate.
Definition 3.1: Proof: An Informal Definition
A proof is a demonstration of the validity of some precise mathematical statement. The demonstration should contain sufficient detail to convince the intended audience of its validity.
Undefined terms
Ideas that have enough intuitive appeal that we may safely use them as a starting place.
Postulates
Axioms with properties that the undefined terms need to satisfy.
Mathematical proof
A mathematical proof of the statement S is a sequence of logically valid statements that connect axioms, definitions, and other already validated statements into a demonstration of the correctness of S. The rules of logic and the axioms are agreed on ahead of time. At a minimum, the axioms should be independent and consistent. The amount of detail presented should be appropriate for the intended audience.
Group
Let G be nonempty set, together with a binary operation, . G is a group if the following four axioms are satisfied.
Closure: For all a, b, G, a b G.
Associativity: For all a, b, c G, a (b c) = (a b) c.
Identity: There is an element e G, called the identity, such that a e = e a = a for all a G.
Inverses: For each element a G, there is an element b G, called the inverse of a, such that a b = b a = e.
The Natural numbers
The set of natural numbers is denoted by N, and is defined by
N = {0, 1, 2, 3, 4,...}
The Integers
The set of integers is denoted by Z, and is defined by
Z = {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}
The Rational Numbers
The set of rational numbers is denoted by Q, and is defined by
Q = { p/q | p Z, q Z, and q = 0}
Irrational Numbers
A real number that is not rational is called irrational.
Divisible
The integer a is divisible by the nonzero integer b if a = bc for some integer c. We denote this by b | a and also say that b divides a.
The Quotient-Remainder Theorem
Let a and b be integers with b = 0. then there exist unique integers q and r such that a = bq and 0 < r < |b|.
Even and Odd
An integer, n, is even if there exists an integer, k, such that n - 2k. An integer is odd if it is not even.
Greatest Common Divisor
Let a and b be integers that are not both 0. The greatest common divisor (gcd) of a and b is a positive integer d such that
- d | a and d | b
- If c divides both a and b, then c | d.
The greatest common divisor of a and b is denoted by gcd(a, b). An alternative notation is (a, b).
gcd(a, b) = as + bt
Let a and b be integers such that at least one is not 0. then there are integers, s and t, such that gcd(a, b) = as + bt.
Least Common Multiple
The least common multiple (lcm) of two integers a and b is a nonnegative integer, m, such that
- a | m and b | m.
- If both a and b divide c, then m | c.
The least common multiple of a and b is denoted by lcm(a, b). An alternative notation is [a, b].
Prime, Composite
A positive integer p, with p > 1, is said to be prime if its only positive integer divisors are 1 and p. A positive integer n, n > 1, that is not prime is called composite. The integer 1 is neither prime nor composite. (In more advanced contexts it is called a unit.)
The Fundamental Theorem of Arithmetic
Every integer, n, with n > 2, can be uniquely written as a product of primes in ascending order.
Relatively Prime
Positive integers a and b are relatively prime if gcd(a, b) = 1.
Pythagorean Triple
The set of integers {a, b, c} is called a Pythagorean triple if a + b = c. it is called a primitive Pythagorean triple if a, b, and c have no common prime factor.
The Well-Ordering Principle
Every nonempty set of natural numbers has a smallest element.
a mod m
Let m be a positive integer. Then a mod m is the remainder when a is divided by m. The integer m is called the modulus.
a b (mod m)
We say that a is congruent to b, mod m, if m divides a - b. This is often expressed as: a b (mod m) if and only if there is an integer k for which a - b = km.
n!
n! = n x (n - 1) x (n - 2) ... 3 x 2 x 1, for positive integers n and is pronounced "n factorial."
Also, 0! = 1 by definition.
Floor Function; Ceiling Function
The floor and ceiling functions are defined for all real numbers x by
floor(x) = x = the largest integer in the interval (x - 1, x]
and
ceiling(x) = x = the largest integer in the interval [x, x + 1)
Proposition 3.22
Let a, b, c, d, m Z with m > 0. If a b (mod m) and c d (mod m) then
1. a + c b + d (mod m).
2. ac bd (mod m).
Proposition 3.23
Let a, k, m Z with k > 0 and m > 1. Then a (a mod m) (mod m).
Proposition 3.24
Let a, b, m Z with m > 0. Then
1. ((a mod m) + (b mod m)) mod m = (a + b) mod m.
2. ((a mod m) x (b mod m)) = ab mod m.
Proposition 3.25: Composition of mod
Let m > k > 1 be two integers. Then (x mod m) mod k = x mod k if and only if k | m.
Proposition 3.26: Cancelation with mod
Let a, b, c, m Z with m ? 1. If gcd(c, m) = 1 and ac bc (mod m), then a b (mod m).
Proposition 3.27
Let a, b, m, n Z with m, n > 1. If a b (mod mn) then a b (mod m) and a b (mod n).
Proposition 3.28
For all real numbers x, 2x < x + x < 2x.
Proposition 3.29
Let a, b, and c be integers, with a = 0. If a | b and a | c, then a | (b + c).
Proposition 3.30
Let a, b, and c be any integers, with a = 0. If a | b, then a | (bc).
Proposition 3.31
Let a, b, and c be integers with a = 0 and b = 0. If a | b and b | c, then a | c.
Proposition 3.32
If n is a positive composite number, then n has at least one prime factor p with 1 < p < n.
Corollary 3.33: Corollary to Proposition 3.32
If a positive integer p > 1 has no divisor d with 1 < d < p, then p is prime.
Proposition 3.34
If the integer n is not even, then n is not even.
Proposition 3.35
The number 2 is irrational.
Proposition 3.36
If x and y are real numbers with x < y, then there exists a real number z with x < z < y.
Proposition 3.37
Let a and b be any two distinct real numbers. Then the following are equivalent:
1. a < b
2. a < (a + b) / 2
3. (a + b) / 2 < b
The Infinitude of the Primes
There is an infinite number of distinct primes.
Proposition 3.39
Let p b a prime. If p divides the product a1a2...an, then p divides at least one of the factors ai.
max; min
Let a and b be real numbers. Then
max(a, b) = a if a > b and b if a < b.
min(a, b) = a if a < b and b if a > b.
Inverses mod m
Let a, m Z with m > 1. Then
11. if gcd (a, m) > 1, a has no inverse, mod m.
2. if gcd(a, m) = 1 then a has an inverse mod m.
3. if a and a are both inverses for a mod m, then a a (mod m).
Corollary 3.45: A Sufficient Condition for a Solution to a Linear Congruence
Let a, b, m Z with m > 1. An expression with variable x of the form ax b(mod m) has an integer solution for x if a and m are relatively prime.
Decomposing a Linear Congruence
Let m > 1 be an integer with factorization m = p p ... p as a product of distinct primes. The linear congruence ax b (mod m) has the same solution set as the system of linear congruences
Fermat's Little Theorem
Let a, p Z with pa prime. Then
a a (mod p)
and
a
Fermat's Last Theorem
Let n Z with n > 2. Then there are no solutions in nonzero integers x, y z to the equation x + y = z .
Lemma 3.51: Euclid's Lemma
Let a, b, c Z with a > 0 and gcd(a, b) = 1. If a | (bc), then a | c.
Proposition 3.52
Let a, b, m1, m2,...mn Z. Suppose mi > 1 for each i and m1, m2,... mn are pairwise relatively prime. if a b (mod mi) for all i, then a b (mod m), where m = m1 x m2 ... mn.
Mathematical Induction
If{P(i)} is a set of statements such that
1. P(1) is true and
2. P(i) ---> P(i + 1) for i > 1
then P(k) is true for all positive integers k. This can be stated more succinctly as
Sum of the First n Positive Integers
for all positive integers, n, with n > 1.
Complete Induction
If {P(i)} is a set of statements such that
1. P(1) is true and
2.
then P(k) is true for all positive integers k. This can be stated more succinctly as
Geometric Progression
A sequence is called a geometric progression if each term in the sequence (after the first) is a constant multiple of the previous term. Thus, if the terms are {ai} for i = 0, 1, 2, 3,..., then ai + 1 = rai for some constant, r.
Arithmetic Progression
A sequence is called an arithmetic progression if each term in the sequence (after the first) is obtained by adding a constant to the previous term. Thus, if the terms are {ai} for i =0, 1, 2, 3,..., then a = ai + d for some constant, d.
Partial Sum of a Geometric Progression
The sum of the first n + 1 elements of a geometric progression depends upon the value of r R.
0^0
In the contexts encountered in this book, 0^0 = 1.
Sum of a Geometric Progression
If r R and |r| < 1, then
Optimal
A stable assignment is called optimal for suitors if every suitor is at least as well off in this assignment as in any other stable assignment.
Possible
A potential mate is called possible for a suitor if there is a stable assignment that pairs them.
Optimality of the Deferred Acceptance Algorithm
The Deferred Acceptance Algorithm produces an assignment that is optimal for every suitor.
The Well-Ordering Principle, Mathematical Induction, and Complete Induction Are Equivalent
The following are equivalent:
-The Well-Ordering Principle (WOP).
-The theorem of mathematical induction (MI).
-The theorem of complete induction (CI).
WOP -----> CI
The well-ordering principle implies the theorem of complete induction.
CI -----> MI
The theorem of complete induction implies the theorem of mathematical induction.
MI -----> WOP
The theorem of mathematical induction implies the well-ordering principle.
Distance in R
Let a = (x1, y1, z1) and b = (x2, y2, z2) be points in R . Then the distance from a to b is
d(a, b) =
Distance in R
Let a = (x1, y1) and b = (x2, y2) be points in R . Then the distance from a to b is
d(a, b) =
...
For any positive integer n, n is either a positive integer, or it is irrational.
Honest
A rational number p / q is honest if and only if p and q have no common factors and q > 1.
Lemma 3.75
If p/q is an honest fraction, then so is (p/q).
Corollary 3.76
If (p/q) is not an honest fraction, then neither is p/q.
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