Question

For any set A prove that: $A\cap A=A$.

Answer 1

Hint: First we will assume that x is a element of set $A\cap A$ , and using this we will prove that $A\cap A$ is a subset of A, and similarly we will take y as a element of A and show that A is a subset of $A\cap A$, and from this two statements we can say that $A\cap A=A$.

Complete step-by-step answer:
Let’s first write the definition terms that will be used,
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
Let us assume that x is an arbitrary element of the set $A\cap A$.
Then, $x\in A\cap A$
$\begin{align}
  & \Rightarrow x\in A\text{ and }x\in A \\
 & \Rightarrow x\in A \\
\end{align}$
Therefore, from this we can say that,
$A\cap A\subseteq A.........(1)$
Now let’s assume that y is an arbitrary element of the set A.
Then, $y\in A$
$\begin{align}
  & \Rightarrow y\in A\text{ and y}\in A \\
 & \Rightarrow y\in A\cap A \\
\end{align}$
Therefore, from this we can say that,
$A\subseteq A\cap A.........(2)$
From (1) and (2) we can say that,
$A\cap A=A$
Hence Proved.

Note: As we have taken some arbitrary elements like x and y, one can take any arbitrary elements they like but the process will be the same. To clarify some doubts one can take some value of A and by using the definition of intersection one can check whether $A\cap A$ equal to A or not.