Question

If A = {a, b, c, d, e}, B = {a, c, e, g} and C= {b, e, f, g}, verify that:
(i) $A\cup B=B\cup A$
(ii) $A\cup C=C\cup A$
(iii) $B\cup C=C\cup B$
(iv) $A\cap B=B\cap A$
(v) $B\cap C=C\cap B$
(vi) $A\cap C=C\cap A$

Answer 1

Hint: Here, we will find the set obtained by each of the given expressions and check whether they hold true or not. The intersection of two sets gives the common element present in them and the union of two sets gives all the elements present in the two sets.

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.
Here, we are given that:
 A = {a, b, c, d, e}
 B = {a, c, e, g}
C= {b, e, f, g}
(i) $A\cup B=B\cup A$
The LHS is:
$A\cup B$ = {a, b, c, d, e, g}
The RHS of the equation is:
$B\cup A$ = {a, c, e, g, b, d}
So, we can see that $A\cup B=B\cup A$.
(ii) $A\cup C=C\cup A$
The LHS of the equation is:
$A\cup C$ ={a, b, c, d, e, f, g}
The RHS of the equation is:
$A\cup C$={ a, b, c, d, e, f, g}
So, we can see that $A\cup C=C\cup A$.
(iii) $B\cup C=C\cup B$
The LHS of the equation is:
$B\cup C$ ={a, c, e, g, b, f}
The RHS of the equation is:
$C\cup B$={ b, e, f, g, a, c}
So, we can see that $B\cup C=C\cup B$.
(iv) $A\cap B=B\cap A$
The LHS of the equation is:
$A\cap B$ ={a, c, e}
The RHS of the equation is:
$B\cap A$={ a, c, e}
So, we can see that $A\cap B=B\cap A$.
(v) $B\cap C=C\cap B$
The LHS of the equation is:
$B\cap C$ ={ e, g}
The RHS of the equation is:
$C\cap B$={ e, g}
So, we can see that $B\cap C=C\cap B$.
(vi) $A\cap C=C\cap A$
The LHS of the equation is:
$A\cap C$ ={ b, e}
The RHS of the equation is:
$C\cap A$={ b, e }
So, we can see that $A\cap C=C\cap A$.
Hence, all the conditions are verified.

Note: Students should note that if an element is present in both the sets of which we are taking union, that element will appear only once in their union. Two sets are said to be equal if and only if all the elements are the same.