Discrete Math

Even Number

A even number is an integer that can be written in the form 2n, where n is an integer

Fallacy(or Contradiction)

A fallacy(or Contradiction) is a statement that is always false.

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.

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.

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.

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

Statement

A statement (proposition) is a sentence that is either true or false, but not both.

Tautology

A tautology is a statment that is always true no matter what truth values are assigned to the statements appearing in it.

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.

Odd Number

An odd number is an integer that can be written in the form 2n+1, where n is an integer

Standard Invalid Arguments

Fallacy of the Converse, Fallacy of the inverse

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

Negations of Quantified Statements

For any P(x) ⌐¥ x P(x)= Ǝ x ⌐P(x) ⌐Ǝ x P(x) = ¥ x ⌐P(x)

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}.

De Morgan's Law

For statements p and q, ⌐(p ᴧ q) ≡ ⌐p v ⌐q ⌐(p v q) ≡ ⌐p ᴧ ⌐q

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.

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),

Test for Primality

If n is a compiste number, then n has a prime divisor that is less than or equal to √n.

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

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

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 }.

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.

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.

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)

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 }.

Domain

Let f : A → B be a function from A to B. We call A the domain of f

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.

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) )

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.

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.

Absorption Laws

P ᴧ ( p v q) ≡ p p v (p ᴧ q) ≡ p

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}.

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┐

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 }.

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

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)

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}.

Universe of Discourse

The set of possible values

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}.

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)

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.

Relatively Prime

Two integers a and b are said to be relatively prime if gcd(a,b) = 1.

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

Euclidean Algorithm

Used to find the greatest common divisor of a and b. It is the last nonzero number.

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.

Negation Laws

p ^ ⌐p ≡ F p v ⌐p ≡ T

Disjunctive Syllogism

p v q ⌐p ---------- Therefore q

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)

Associative Laws

p ᴧ (q ᴧ r) ≡ (p ᴧ q) ᴧ r p v (q v r) ≡ (p v q) v r

Universal Bound Laws

p ᴧ F ≡ F p v T ≡ T

Identity Laws

p ᴧ T ≡ p p v F ≡ p

Impotent Laws

p ᴧ p ≡ p p v p ≡ p

Commutative Laws

p ᴧ q ≡ q ᴧ p p v q ≡ q v p

Modus Ponens

p → q p ---------- Therefore q

Fallacy of the Converse

p → q q ---------- Therefore p

Negation of Conditional

⌐( p → q) ≡ p ᴧ ⌐ q

Double Negation Law

⌐(⌐p) ≡ p

Contingency

A statement that is sometimes false and sometimes true is called a contingency

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.

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.

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.)

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

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.

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.

Variations of the Conditional Statement

Conditional p→ q Converse q → p Inverse ⌐p → ⌐q Contrapositive ⌐q → ⌐p

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.

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|.

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.

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.

Lemma 3.1

If a, b , q and r are integers, and a = bq + r, then gcd(a,b) = gcd(b,r).

Standard Valid Arguments

Modus Ponents, Modus Tollens, Reasoning by Transitivity(Law of Hypothetical Syllogism), Disjunctive Syllogism

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

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.

Empty Set

The empty set is the set with no elements, and it is denoted by Ø. It may also be denoted by { }.

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┘.

Reasoning by Transitivity(Law of Hypothetical Syllogism)

p → q q → r ------------ Therefore p → r

Fallacy of the Inverse

p → q ⌐p ---------- Therefore ⌐q

Modus Tollens

p → q ⌐q ---------- Therefore ⌐p