Prove That B ⊂ A ∪ B
Question:
For any two sets \( A \) and \( B \), prove that:
\[ B \subset A\cup B \]Solution
Let \( x \in B \).
Then \( x \) belongs to set \( B \).
By the definition of union of sets, if an element belongs to at least one of the sets \( A \) or \( B \), then it belongs to \( A\cup B \).
Therefore,
\[ x\in A\cup B \]Hence every element of \( B \) is also an element of \( A\cup B \).
Therefore,
\[ B\subset A\cup B \]Hence proved.