# Universal, disjoint sets | Complement of a set

The universal set. In some problems involving sets, it is necessary to consider one or more sets under consideration as belonging to some larger set that contains them. For instance, if we were considering the set of math teachers (say, M) in a school, it might be convenient to the universal set (say, U) as all the teachers in the school. In other words, where a universal set has been defined, all the sets under consideration must necessarily be subsets of it.

The universal set is usually denoted by ∪

The complement of a set. If A is any set, with some universal set U defined, the complement of A, normally written as A’, is the defined as “all those members that are not contained in A but are contained in U”.

Disjoint sets. Sets that have different elements (that is, no elements in common) are called disjoint sets.

Set equality. Two sets are equal if they have identical elements.

If A = {r, s, t} and B = {r, t, s} then A = B

The number of a set. The number of a set A, written as n[A] is defined as the number of elements that A contains. For example:

If A = {a, b, c, d}, then n[A] = 4 (since there are 4 elements in A).

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