Home » Blog » Prove union and intersection of a set with itself equals the set

Prove union and intersection of a set with itself equals the set

Prove that A \cup A = A and A \cap A = A.


Proof. Let x be an arbitrary element of A \cup A. Then x \in A or x \in A; hence, x \in A. Thus, A \cup A \subseteq A.
Conversely, if x is an arbitrary element of A then x \in A \cup A since it is in A. Thus, A \subseteq A \cup A.
Therefore, A \cup A = A. \qquad \blacksquare

Proof. Let x be an arbitrary element of A \cap A. Then x \in A and x \in A; hence, x \in A. So, A \cap A \subseteq A.
Conversely, if x \in A is arbitrary, then x \in A and x \in A; hence, x \in A \cap A. Thus, A \cap A \subseteq A.
Therefore, A \cap A = A. \qquad \blacksquare

2 comments

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):