Prove That A∩B = Φ ⟺ A ⊆ B’
Question:
For any two sets \( A \) and \( B \), prove that:
\[ A\cap B=\phi \iff A\subseteq B’ \]Solution
First, prove that \( A\cap B=\phi \Rightarrow A\subseteq B’ \)
Assume:
\[ A\cap B=\phi \]This means there is no element common to both \( A \) and \( B \).
Let \( x\in A \).
Since \( A\cap B=\phi \), therefore:
\[ x\notin B \]Hence,
\[ x\in B’ \]Therefore every element of \( A \) belongs to \( B’ \).
\[ A\subseteq B’ \]Now, prove that \( A\subseteq B’ \Rightarrow A\cap B=\phi \)
Assume:
\[ A\subseteq B’ \]Then every element of \( A \) belongs to \( B’ \).
So no element of \( A \) belongs to \( B \).
Hence there is no common element between \( A \) and \( B \).
\[ A\cap B=\phi \]Therefore,
\[ A\cap B=\phi \iff A\subseteq B’ \]Hence proved.