Question

For any set A prove that: $A\cap U=A$ , where U is the universal set.

Answer 1

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

Complete step-by-step answer:
Let’s first write the definition of some important terms.
Universal set: The set containing all objects or elements and of which all other sets are subsets.
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 $A\cap U$.
Then, $x\in A\cap U$
$\begin{align}
  & \Rightarrow x\in A\text{ and }x\in U \\
 & \Rightarrow x\in A \\
\end{align}$
Therefore, from this we can say that,
$A\cap U\subseteq A.........(1)$
Now let’s assume that y is an arbitrary element of the set A.
Then, $y\in A$
As we know that $A\subset U$ , then x must be an element of U also.
Hence,
$\begin{align}
  & \Rightarrow y\in A\text{ and y}\in U \\
 & \Rightarrow y\in A\cap U \\
\end{align}$
Therefore, from this we can say that,
$A\subseteq A\cap U.........(2)$
From (1) and (2) we can say that,
$A\cap U=A$
Hence Proved.

Note: In this we have taken some arbitrary element x and y, one can take any value they like and solve using the same process we will get the same answer. One can also take some value of A and U, and then by performing the required operations to check the relation $A\cap U=A$.