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Definitions from semeser
Terms in this set (82)
Statement
A statement (proposition) is a sentence that is either true or false, but not both.
Negation
A negation of a statement p is the statement "not p" or "it is not the case that p," and is denoted by ⌐p. A statement and its negation always have the opposite truth value.
Conjunction
The conjunction of two statements p and q is the statement "p and q," and is denoted by p ᴧ q. The conjunction p ᴧ q is true if both p and q are true and is false otherwise.
Disjunction
The disjunction of two statements p and q is the statement "p or q," and is denoted by p v q. The disjunction p v q is true if either of p and q are true or if both are true.
Conditional
A statement of the form "if p, then q" where p and q are statements, is called a conditional and is denoted by p -> q.
Forms of the conditional statement
The conditional, p→ q can be stated in any of the following ways:
If p, then q
q if p
p implies q
p only if q
p is sufficient for q
q is necessary for p
Tautology
A tautology is a statment that is always true no matter what truth values are assigned to the statements appearing in it.
Fallacy(or Contradiction)
A fallacy(or Contradiction) is a statement that is always false.
Contingency
A statement that is sometimes false and sometimes true is called a contingency
De Morgan's Law
For statements p and q,
⌐(p ᴧ q) ≡ ⌐p v ⌐q
⌐(p v q) ≡ ⌐p ᴧ ⌐q
Commutative Laws
p ᴧ q ≡ q ᴧ p
p v q ≡ q v p
Associative Laws
p ᴧ (q ᴧ r) ≡ (p ᴧ q) ᴧ r
p v (q v r) ≡ (p v q) v r
Distributive Laws
p ᴧ (q v r) ≡ (p ᴧ q) v (p ᴧ r)
p v (q ᴧ r) ≡ (p v q) ᴧ (p v r)
Impotent Laws
p ᴧ p ≡ p
p v p ≡ p
Double Negation Law
⌐(⌐p) ≡ p
Negation Laws
p ^ ⌐p ≡ F
p v ⌐p ≡ T
Identity Laws
p ᴧ T ≡ p
p v F ≡ p
Universal Bound Laws
p ᴧ F ≡ F
p v T ≡ T
Absorption Laws
P ᴧ ( p v q) ≡ p
p v (p ᴧ q) ≡ p
Variations of the Conditional Statement
Conditional p→ q
Converse q → p
Inverse ⌐p → ⌐q
Contrapositive ⌐q → ⌐p
Negation of Conditional
⌐( p → q) ≡ p ᴧ ⌐ q
Universe of Discourse
The set of possible values
Universal Quantification
The universal quantification of P(x) is the statement " For all x in the universe of discourse, P(x) is true." It is denoted by
¥ x P(x)
Existential Quantification
The existential quantification of P(x) is the statement "There exists an x in the universe of discourse for which P(x) is true." It is denoted by Ǝ x P(x)
Negations of Quantified Statements
For any P(x)
⌐¥ x P(x)= Ǝ x ⌐P(x)
⌐Ǝ x P(x) = ¥ x ⌐P(x)
Argument
An argument is a series of statements, called PREMISES, followed by another statement, called a CONCLUSION. An argument is called VALID if the conclusion is true whenever all the premises are true.
Standard Valid Arguments
Modus Ponents, Modus Tollens, Reasoning by Transitivity(Law of Hypothetical Syllogism), Disjunctive Syllogism
Standard Invalid Arguments
Fallacy of the Converse, Fallacy of the inverse
Modus Ponens
p → q
p
----------
Therefore q
Modus Tollens
p → q
⌐q
----------
Therefore ⌐p
Reasoning by Transitivity(Law of Hypothetical Syllogism)
p → q
q → r
------------
Therefore p → r
Disjunctive Syllogism
p v q
⌐p
----------
Therefore q
Fallacy of the Converse
p → q
q
----------
Therefore p
Fallacy of the Inverse
p → q
⌐p
----------
Therefore ⌐q
Even Number
A even number is an integer that can be written in the form 2n, where n is an integer
Odd Number
An odd number is an integer that can be written in the form 2n+1, where n is an integer
Rational Number
A rational number is a number that can be written in the form a/b, where a and b are integers and
b≠0
Indirect Proof
To prove a statement of the form p → q, it is sometimes easier to prove the logically equivalent contrapositive ⌐q → ⌐p.
Contradiction
To prove a statement p, start by assuming the statement is false. In other words assume ⌐p. Then show that assuming ⌐p leads to a contradiction.
Set
A set is an unordered collection of objects. The objects in the set are called the ELEMENTS of the set. If x is an element of a set S, we denote this by x Є S. If x is not an element of a set S, we denote this by x Ɇ S.
Empty Set
The empty set is the set with no elements, and it is denoted by Ø. It may also be denoted by { }.
Equal Sets
Two sets A and B are said to be equal if they have the same elements. In this case we write A = B
Subset
A set B is said to be a subset of a set A if every element of B is also an element of A. In other words, x Є B implies x Є A. To denote that B is a subset of A, we write B _C_ A
Cardinality
If A is the finite set, then the cardinality of A is the number of distinct elements in A, and is denoted by |A|.
Cartesian Product
The Cartesian Product of two sets A and B is the set of all ordered pairs (a,b), where a Є A and
b Є B. Is denoted by A x B. Therefore,
A x B = { (a,b) | a Є A and b Є B}.
Power Set
For any set S, the power set of S is the set whose elements are all the subsets of S, and it is denoted by P(S). in other words,
P(S) = {B | B c S}.
Union
The union of two sets A and B is the set whose elements are elements of A, B, or both. It is denoted by A U B. Therefore,
A U B= {x | x Є A or x Є B}.
Intersection
The intersection of two sets A and B is the set whose elements are elements of both A and B. It is denoted by A ∩ B. Therefore,
A ∩ B = { x | x Є A and x Є B}.
Difference
Let A and B be sets. The diffence of A and B is the set whose elements are elements of A but not elements of B. It is denoted by A - B. Therefore,
A - B = { x |x Є A and x Є B }.
Complement
The complement of a set A is the set U - A whose elements are the elements of the universal set U that are not elements of A. It is denoted by Ā. Therefore,
Ā = { x | x Ɇ A }.
Divides
If a and b are integers and there is an integer c such that b = a • c, then we say a divides b or b is divisible by a, and write a|b. In this case, we say that a is a factor or divisor of b and that b is a multiple of a. If a does not divide b, we write a ł b.
Properties of Divisibility
For all integers a, b and c, the following properties hold.
1) If a | b and b | c, then a | c
2) if a | b and a | c, then a | ( b + c)
3) if a | b, then a | bc
Corollary 3.1
If a, b and c are integers, and a | b and a | c, then for any integers m and n, a | ( mb + nc),
Prime Number
A prime Number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer greater than 1 that is not prime is called a composite number.
Test for Primality
If n is a compiste number, then n has a prime divisor that is less than or equal to √n.
The Division Algorithm
Given any integer a and positive integer d, there exists unique integers q and r such that
a = dq + r with 0 ≤ r < d.
Greatest Common Divisor
Let a and b be integers that are not both 0. The greatest common divisor of a and b, denoted gcd(a,b), is the largest integer d such that d | a and d | b.
Relatively Prime
Two integers a and b are said to be relatively prime if gcd(a,b) = 1.
Lemma 3.1
If a, b , q and r are integers, and a = bq + r, then gcd(a,b) = gcd(b,r).
Euclidean Algorithm
Used to find the greatest common divisor of a and b. It is the last nonzero number.
Theorem 3.6
If a and b are positive integers, then gcd(a,b) can be written as an integral linear combination of a and b. in other words, tehre exists integers s and t such that gcd(a,b) = sa + tb.
a modulo n (a mod n)
If n is a positive integer and a is any integer, then the remainder we get when using the division algorithm to divide a by n is calld a modulo n. and is denoted a mod n
Congruent modulo n
Let n be a positive integer. Two integers a and b are said to be congruent modulo n if n | (b - a). We write a ≡ b (mod n) to denote a is congruent to b modulo n, and write a ≠ b (mod n) to indicate that this is not the case.
Congruence Modulo n.
Let n be a positive integer, and a and b any integers. Then a ≡ b (mod n) if and only if the leave the same remainder when divided by n using the division algorithm, that is if and only if
a mod n = b mod n.
Properties of Modular Arithmetic
Let a, b, c, d, and n be integers with n > 1, and suppose a ≡ b (mod n) and c ≡ d (mod n). Then,
1) (a + c) ≡ (b + d) ( mod n)
2) (a - c) ≡ ( b - d) ( mod n)
3) ac ≡ bd (mod n)
Inverse of a modulo n.
If a is an integer and ā a ≡ 1( mod n), then we say ā is an inverse of a modulo n.
Chinese Remainder Theorem
Suppose n1,n2,....,nk are integers which are pairwise relatively prime. Then for any integers a1, a2,...,ak, there is a solution x to the system of equations
x ≡ a1 (mod n1)
x≡ a2(mod n2).......
x≡ ak(mod nk)
x = a1• N1•x1 + a2•N1•xd mod N
where,
N = n1•n2
N1= N/n1
N2=N/n2
x1= inverse of N1 mod n1
x2 = inverse of N2 mod n2
Function
Let A and B be sets. A function f from A to B associates each element of A with a unique element of B. More formally, ever a Є A is associated with a unique element f(a) = b Є B called the image of a under f. We say that f MAPS a to f(a). This function sometimes denoted by
f: A → B
Domain
Let f : A → B be a function from A to B. We call A the domain of f
Codomain
Let f : A → B be a function from A to B. The set B is called the codomain of f.
Range
Let f : A → B be a function from A to B. The set of all images of the elements of A under the function f is called the range of f. in other words
range of f = { b Є B | b = f(a) for some a Є A }.
one-to-one
A function f : A → B from A to B is called one-to-one if
f (a1) = f (a2) implies a1 = a2 for all a1, a2 Є A.
Onto
A function f : A → B from A to B is called onto if every element of B is an image of some element of A under f. That is, f is onto if given any b Є B, there exists some element a Є A such that
f (a) = b
one-to-one correspondence (bijection)
A function f is called one-to-one correspondence (bijection) if it is both one-to-one and onto.
Composition
Let g: A → B be a function from the set A to the set B, and let f : B → C be a function for the set B to the set C. Then the composition of f and g is the function f ᵒ g from A to C defined by
( f ᵒ g ) (a) = f ( g (a) )
Inverse Function
Let f : A → B be a one-to-one correspondence from the set A to the set B. Then the inverse function of f is the function f^-1: B → A defined by,
f ^ -1 (b) = a, where a is the unique element of A for which f(a) = b.
Floor Function
The floor function is a function from R to Z that maps x to the largest integer that is less than or equal to x. It is denoted by └x┘.
Ceiling Function
The ceiling function from R to Z maps x to the smallest integer that is greater than or equal to x. It is denoted by ┌x┐
The Principle of Mathematical Induction
Let P(n) be a statement that is defined for all integers n greater than or equal to a, where a is some fixed integer. Suppose that the following two statements are true:
1) P(a) is true.
2) For all integers k ≥ a, if P(k) is true, then
P(k + 1) is true.
Then P(n) is true for all integers n ≥ a.
Cardinality
If A and B are 2 sets, then we say A and B have THE SAME cardinality if there exists a map
f : A → B that is a bijection.
Larger Cardinality
We say that set A has LARGER cardinality than B if there exists a one-to-one function g : B → A. and there does not exists a bijection h : B → A.
Countable
A set A is called countable(countably infinite) if there exists a bijection f : { 1,2,3,....} → A
(i.e. - we can make an infinite lists of all elements of A.)
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