Question

Let A and B be sets. Show that$A\subseteq B$ if and only if$A\cap B=A$

Short answer

$A\cap B=A$

Step-by-step solution

Step: 1

$$\begin{gathered} x\in A\cap B \\ \Rightarrow x\in Aandx\in B \\ \Rightarrow A\cap B\subseteq A \end{gathered}$$

Step: 2

$$\begin{gathered} x\in A \\ \because A\subseteq B \\ x\in B \\ \Rightarrow x\in A\cap B \\ A\subseteq A\cap B \end{gathered}$$

Hence$A\cap B=A$