Complement of a set (A′)

How to read it

A complement only makes sense against a universal set U, the “everything” under discussion. OpenStax’s Contemporary Mathematics defines A′ = {x ∈ U | x ∉ A}: if U is the digits 0 to 9 and A the primes {2, 3, 5, 7}, then A′ = {0, 1, 4, 6, 8, 9}. Oscar Levin’s Discrete Mathematics defines the complement the same way, as everything not an element of A. In probability the same mark is used for an event: OpenStax’s Introductory Statistics says A′ consists of all outcomes not in A, and that P(A) + P(A′) = 1.

Also searched as: A′, A bar, A^c, Aᶜ, A prime, complement symbol, set complement.

A′, Aᶜ or Ā

The notation varies by book. OpenStax writes A′ and notes that the Sets Challenge app it recommends writes Aᶜ (A with a small c) for the same thing. Levin writes the complement with a bar over the letter, Ā. Unicode also has a dedicated character, ∁ COMPLEMENT (U+2201). All four mean the same set.

Look-alikes

In statistics, a bar over a letter means something else: x̄ is a sample mean. The complement is also close to set difference: A′ is U ∖ A, everything in the universal set except A.