Empty, finite and equal sets

The empty set (also called the null or void set) has no elements at all. It is written φ or { }.

Example: {x : x ∈ N and x = 2x + 3}. Solving gives x = –3, which is not a natural number, so no element qualifies: the set is φ.

Do not mix up three different things: φ is a set with nothing in it; {0} is a set with one element, the number 0; and 0 on its own is a number, not a set at all.

A set with exactly one element is a singleton: {x : x + 4 = 9} = {5}.

n(A), the cardinal number of A, is the number of distinct elements in A.

• A set is finite if it is empty or n(A) is a definite number: the students of your school, or even all the animals on the earth (huge, but countable in principle).
• Otherwise it is infinite: N, the multiples of 50 {50, 100, 150, ...}, the points on a line.

Two sets are equal (A = B) if they have exactly the same elements. Order and repetition do not matter: {3, 4} = {4, 3, 3}.

To compare sets given by rules, first list their elements. {x : x² – 7x + 12 = 0} = {3, 4}, because x² – 7x + 12 = (x – 3)(x – 4). So it equals {4, 3}.

Sets with the same number of elements but different elements are only equivalent, not equal. {a, b, c} and {1, 2, 3} are equivalent (n = 3 each) but not equal. Equal sets are always equivalent; equivalent sets need not be equal.