Question

For any two sets A and B , in a universal set U , prove that
A ⊆ B ⇔ A ∪ B = B.

Answer 1

Answer on Question #78507 - Math - Discrete Mathematics

June 26, 2018

Question. For any two sets $A$ and $B$, in a universal set $U$, prove that

$A\subseteq B\iff A\cup B=B.$

Answer. Assume that $A$ and $B$ are sets in a universal set $U$.

First, we will prove the implication from left to right. Assume that $A\subseteq B$. We need to prove $A\cup B=B$ which is equivalent to

$x\in A\cup B\iff x\in B$

for every $x\in U$.

- (From left to right.) Let $x\in A\cup B$. Then $x\in A$ or $x\in B$ by the definition of union.

- Assume $x\in A$. Then $x\in B$ by the assumption $A\subseteq B$.

- Assume $x\in B$. Then $x\in B$.

Hence in both cases $x\in B$.

- (From right to left.) If $x\in B$, then $x\in A\cup B$ by the definition of union.

Second, we will prove the implication from right to left. Assume that $A\cup B=B$. We need to prove $A\subseteq B$. Let $x\in A$. Then $x\in A\cup B$ by the definition of union. From the assumption $A\cup B=B$, it follows that $x\in A\cup B$ implies $x\in B$ for every $x\in U$. Hence $x\in B$.