Complement

Complement

  • The complement of a set is defined when our given set is understood to be a subset of some much larger set called the universe or universal set.

  • When \(X\) is a set and its universal set \(U\) is specified (or understood) then the complement \(\overline{X}\) is the set \(U-X\).

Example

  • \(P\) is the set of prime numbers, with universal set \(U=\mathbb{N}\). What is \(\overline{P}\)?

Example

\(X=(1,3)\times [1,2]\) in \(\mathbb{R}^{2}\), with universal set \(U=\mathbb{R}^{2}\). Sketch \(\overline{X}.\)

Example

Suppose:

  • \(A=\{x : x\in\mathbb{N}, x\mathrm{\ is\ even\ and\ } 0\le x\le 8\}\)
  • \(B=\{x : x\in\mathbb{N}, x\mathrm{\ is\ odd\ and\ }0\le x\le 8\}\)
  • \(U=\{x : x\in\mathbb{N}, 0\le x\le 8\}\).

What is \(\overline{A}\cap B\)?

Example

\(X=\{(x,y)\in\mathbb{R}^{2} : y<x^2\}\) with universal set \(\mathbb{R}^{2}\). Sketch \(\overline{X}\).