Question

For three sets A, B and C, show that $A\cap B=A\cap C$ need not to imply B = C.

Answer 1

Hint: Here, we will consider three different sets A, and C and find $A\cap B$ and $A\cap C$. We will consider the sets such that $A\cap B=A\cap C$. After that we will check whether B and C are equal or not.

Complete step-by-step answer:
A set is a well defined collection of distinct objects, considered as an object in its own right. The union of a collection of a set 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. The union of two sets A and B is the set of elements which are in A, in B or in both A and B. It is denoted as $A\cup B$. The intersection of two sets A and B denoted by $A\cap B$ is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. So, x is said to be an element of this intersection $A\cap B$ if and only if x is an element of both A and B.
Consider the set A as:
A = {a, b, c, d, e}
Then, let us consider the sets B and C as:
B = {1, b, c, 2, f} and C = {3, 5, g, b, c}
The set $A\cap B$ is given as:
$A\cap B=\{b,c\}$
Similarly, the set $A\cap C$ is given as:
$A\cap C=\{b,c\}$
Here, we observe that \[A\cap B=A\cap C=\{b,c\}\], but $B\ne C$.
Hence, it is proved that if $A\cap B=A\cap C$, then it is not necessary that B = C.

Note: Students should note here that two sets are said to be equal to each other if and only if all the elements of both the sets are the same. None of the elements of the sets should be different.