Question

Prove that for all sets A and B:
A⊆B ⇐⇒ A∪B=B.

Answer 1

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Answer on Question #76358 – Math – Discrete Mathematics

Question

Prove that for all sets A and B:

$$A \subseteq B \Leftrightarrow A \cup B = B$$

Solution

1) $A \subseteq B \Rightarrow A \cup B = B$.

Assume that $A \subseteq B \Rightarrow A \cup B \neq B$ . As all elements from $B$ are contained in $A \cup B$ , there exists $x$ , such as $\left\{ \begin{array}{l} x \in A \cup B \\ x \notin B \end{array} \right.$ . Consequently $\left\{ \begin{array}{l} x \in A \\ x \notin B \end{array} \right.$ . But using that fact and $A \subseteq B$ one gets a contradiction. Thus, $A \subseteq B \Rightarrow A \cup B = B$ holds true.

2) $A \subseteq B \Leftarrow A \cup B = B$ . If not, there exists an element $x$ from $A$ that isn't contained in $B$ . Consequently $x \in A \cup B$ which is equal to $B$ and it comes to contradiction.

Thus, $A \subseteq B \Leftarrow A \cup B = B$ holds true.

It follows from 1) and 2) that $A \subseteq B \Leftrightarrow A \cup B = B$ .