Question

If A = {a, b, c, d, e}, B ={a, c, e,g} and C={b, e, f, g}, verify that:
$\left( A\cup B \right)\cup C=A\cup \left( B\cup C \right)$ and $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$

Answer 1

Hint : Here, we will find the sets obtained on doing the operations given in the problem and then verify whether the given equations are true. The union of two sets consists of all the elements belonging to both the sets and the intersection of two sets consist of elements which are common to both of them.

Complete step by step solution :
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.
Here, we are given:
A= {a, b, c, d, e}
B ={a, c, e,g}
C={b, e, f, g}
So we can write:
$A\cup B$ consists of all the elements which belong to both A and B.
So, $A\cup B=\\{a,b,c,d,e,g\\}............\left( 1 \right)$
$A\cap B$ consists of elements which belong to both A and B.
So,$A\cap B=\\{a,c,e\\}........\left( 2 \right)$
Similarly $B\cup C$ comprises of all the elements belonging to B and C.
So, $B\cup C=\\{a,b,c,e,f,g\\}............\left( 3 \right)$
And $B\cap C$ consist of all elements that are common to B and C.
So, $B\cap C=\\{e,g\\}..........\left( 4 \right)$
(i) $\left( A\cup B \right)\cup C=A\cup \left( B\cup C \right)$
Here, LHS is:
$\left( A\cup B \right)\cup C$
Since, $A\cup B=\\{a,b,c,d,e,g\\}$ (from equation 1) .
$\left( A\cup B \right)\cup C$ consists of all elements belonging to $A\cup B$ and C.
Therefore, $\left( A\cup B \right)\cup C=\\{a,b,c,d,e,f,g\\}$
And, RHS is:
$A\cup \left( B\cup C \right)$
From equation (3), we have:
$B\cup C=\\{a,b,c,e,f,g\\}$
$A\cup \left( B\cup C \right)$ consists of all elements belonging to both A and $B\cup C$.
So, $A\cup \left( B\cup C \right)=\\{a,b,c,d,e,f,g\\}$
Since, LHS=RHS.
So, $\left( A\cup B \right)\cup C=A\cup \left( B\cup C \right)$.
(ii) $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$
Here, LHS is:
$\left( A\cap B \right)\cap C$
From equation (2), we have:
$A\cap B=\\{a,c,e\\}$
$\left( A\cap B \right)\cap C$ consists of all elements which are common to $A\cap B$ and C.
Since, the only element which exists in both $A\cap B$ and C is ‘e’.
Therefore, $\left( A\cap B \right)\cap C=\\{e\\}$
And, RHS is:
$A\cap \left( B\cap C \right)$
From, equation (4), we have:
$B\cap C=\\{e,g\\}$
Now, $A\cap \left( B\cap C \right)$ consists of all elements which are common to A and$B\cap C$. The only element which exists both in A and $B\cap C$ is ’e’.
So, $A\cap \left( B\cap C \right)=\\{e\\}$
Since, LHS=RHS.
So, $\left( A\cap B \right)\cap C=A\cap \left( B\cap C \right)$.
Hence, both equations are verified.

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 same. None of the elements of the sets should be different. Students should find the sets involving intersection and union correctly.