Question

Given $\xi $ = (a, b , c, d, e, f, g ,h ,I, j, k) , C= (b, e, a, d) , D = (j, a ,d , e) and E = (b, a ,d, g, e). Determine,
A. $C \cap D \cap E$
B. $C \cup D \cup E$
C. n[($C \cup D$) $ \cap E$ ]
D. n[($C \cup E$) $ \cap D$ ]

Answer 1

Hint:This question is based on set theory rules. Apply suitable operations to get intersection or union as required operations on the sets. Also, n(A) will be the number of elements in the set A.

Complete step-by-step answer:
Given data in the question,
Given $\xi $ = (a, b , c, d, e, f, g ,h ,I, j, k) , C= (b, e, a, d) , D = (j, a ,d , e) and E = (b, a ,d, g, e).
Here $\xi $ is the universal set.

A. $C \cap D \cap E$
It will be all common elements from the 3 sets. So we have
$\therefore $ $C \cap D \cap E$ will be = ( a, d, e )
B.$C \cup D \cup E$
It will be all elements present in at least one set. So we have,
 $\therefore $ $C \cup D \cup E$ will be = (b, e , a, d , j, g)
C.n[($C \cup D$) $ \cap E$ ]
First we find
$C \cup D$ = (b, e, a, d, j)
Then we get ($C \cup D$) $ \cap E$ as common from $C \cup D$ and E ,
($C \cup D$) $ \cap E$ = (b, e, a, d)
Thus the number of elements in this set will be 4.
$\therefore $ n[($C \cup D$) $ \cap E$ ] = 4
D.n[($C \cup E$) $ \cap D$]
First we find
$C \cup E$ = (b, e, a, d, g)
Then we get ($C \cup E$) $ \cap D$ as common from $C \cup E$ and D ,
($C \cup E$) $ \cap D$ = (a, d, e)
Thus the number of elements in this set will be 3.
$\therefore $ n[($C \cup E$) $ \cap D$] = 3

Note:Set theory is really an interesting branch of algebra. Various set related operations like intersection, union, compliment, difference etc. are very common operations. Many real life problems are solvable by using set theory rules properly.