CIS1600 Final

odd integer

An integer is odd iff n = 2k + 1 for some integer k

divisible by

An integer n being divisible by an integer k is denoted by k | n

even integer

An integer n is even iff n = 2k for some integer k

prime integer

An integer n is prime iff n > 1 and for all positive integers r and s, if n = r, then r = 1 or s = 1. Otherwise, n is composite.

inverse

The implication ¬p ⇒ ¬q is called the inverse of p ⇒ q

contrapositive

The implication ¬q ⇒ ¬p is the contrapositive of p ⇒ q

proposition

a statement that is either true or false

negation

denoted as ¬p, is the proposition that is true when p is false and vice-versa

prime factorization theorem

every positive integer can be uniquely represented as a product of primes

converse

implication q ⇒ p is called the converse of the implication p ⇒ q

necessary condition

p is a necessary condition for q means that ¬p ⇒ ¬q, or equivalently q ⇒ p

sufficient condition

p is a sufficient condition for q means p ⇒ q

biconditional

p ⇐⇒ q is the proposition that is true if p and q have the same truth values and is false otherwise

Implication

p ⇒ q is the proposition that is false when p is true and q is false and true otherwise

conjunction

p ∧ q is the proposition that is true when both p and q are true

disjunction

p ∨ q is the proposition that is true when at least one of p or q is true

Exclusive or

p ⊕ q is the proposition that is true when exactly one of p and q is true, false otherwise