Question

Show that the closure of a set equals the union of the set and its complement.

Answer 1

Definition of complement. Given a set A, the complement of A is the set of all elements in the universal set X, but not in A. We can write $A^c$

Definition of closure: if X is a set and A is a subset of X, then the **closure** of A is the intersection of all closed sets in X containing A, i.e. the smallest closed set in X containing A.

In the context of the task it appears that by closure of the set the universal set is meant.

Let X be a set and A,B are its subsets.

Then $A \cup A^C = X$.

Proof.

$a \in A \cup A^C$ iff $a \in A$ or $a \in A^C$ iff $a \in A$ or $a \notin A$ iff $a \in X$ □