Discrete Mathematics Definitions - Exam 1

If a and b are integers, a divides b

if there exists an integer k so that b = ak

For a natural number n, n! =

n(n−1)(n−2)···3·2·1.

An integer n is even

if there exists an integer k so that it can be written as n = 2k

Two statements are equivalent

if their truth table match for all possible values of the component statements

Let a and b be integers. The greatest common divisor of a and b is denoted by gcd(a,b) and is the natural number d that satisfies the following two conditions

d divides both a and b. If n is an integer that divides both a and b then n divides d.

natural number definition

a counting number as in 1, 2, 3

whole number definition

a counting number or zero as in 0, 1, 2, 3

integer definition

a positive or negative counting number or zero

An integer n is odd

if there exists an integer k so that it can be written as n = 2k +1

mathematical statement definition

a declarative sentence which is either true or false.

negation definition

a statement having the opposite truth value of the original statement.

A conditional statement is

one of the form If P then Q

The converse of the conditional statement "If P, then Q" is

If Q, then P

The inverse of the conditional statement "If P, then Q" is

If not P, then not Q

The contrapositive of the conditional statement "If P, then Q" is

If not Q, then not P

A number n is a rational number if

it can be written as p/q for some integers p and q where q does not equal 0.

A natural number n is prime if

it has exactly two divisors namely 1 and itself