Definitions

Even Integer

An integer n is even if, and only if, n = 2k for some integer k. 

Symbolically,

n is even ↔ ∃ an integer k such that n = 2k.

Odd Integer

An integer n is odd if, and only if, n = 2k + 1 for some integer k.

Symbolically,

n is odd ↔ ∃ an integer k such that n = 2k + 1.

Prime Number

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

Symbolically, if integer n > 1, then

n is prime ↔ ∀ positive integers r and s, if n = r. s then r = 1 or s = 1.

An alternative way to define a prime number is to say that an integer n > 1 is prime if, and only if, its only positive integer divisors are 1 and itself.

Composite Number

An integer n is composite if, and only if, n = r. s for some positive integers r and s with r ≠ 1 and s ≠ 1.

Symbolically, if integer n > 1, then

n is composite ↔ ∃ positive integers r and s such that n = r. s and r ≠ 1 and s ≠ 1.

Rational Number

A real number r is rational if, and only if, r = a/b for some integers a and b with b ≠ 0. A real number that is not rational is irrational.

More formally, if r is a real number, then 

r is a rational ↔ ∃ integers a and b such that r = a/b and b ≠ 0.

Note that the word rational contains the word ratio, which is another word for quotient. A rational number is a fraction or ratio of integers.