Question

Using properties of sets, show that for any two sets A and B $\left( A\cup B \right)\cap \left( A\cap B' \right)=A$.

Answer 1

Hint:At first, try to think about distributive law of set which states that for any three sets A, B, C the statement $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$ then use it for sets A, B, B’ and write the statement $\left( A\cup B \right)\cap \left( A\cap B' \right)$ as $A\cap \left( B\cup B' \right)$ . Now, apply the fact that $B\cup B'$ represents a universal set hence, solve the problem.

Complete step-by-step answer:
In the question, we are asked to show that for any sets A, B; $\left( A\cup B \right)\cap \left( A\cap B' \right)=A$ this statement will hold true.
At first, we briefly understand what is set.
In mathematics sets is a well-defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the number 2, 4, 6 are distinct and considered separately, but they are considered collectively then for mn single set of size three written as $\left\{ 2,4,6 \right\}$ which could also be written as $\left\{ 2,6,4 \right\}$ .
Here Universal set U means the number of elements contained in U is the maximum number of elements any other set can have.
We know one of the distributive law of sets which states that if A, B, C are some sets, they are related as
$A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$
So, here we will consider element A, B, B’.
So, we can write the given statement $\left( A\cup B \right)\cap \left( A\cap B' \right)$ as $A\cap \left( B\cup B' \right)$ .
Here B’ represents all the elements except B. So, $B\cup B'$ means that it represents all the elements of Universal set.
Now A is a set that is contained in a Universal set. So, its intersection will be A itself.
Hence, the statement is proved.

Note: Students generally have confusion between $\left( \cup \right)$ and intersection $\left( \cap \right)$ . If $A\cup B$ is given then it represents the element of both A and B collectively and if $A\cap B$ is given then it represents the elements which are common to both A and B.Students should remember $B\cup B'$ means that it represents all the elements of Universal set.