Answer
(a)
First, perform the operation inside the parentheses of the set\[\left( A\cap B \right)\cup C\].
Now, we compute\[A\cap B\].
Set \[A\cap B\] contains all the elements thatare common to both sets\[A\]and\[B\].
In the provided Venn diagram,regions I, II, IV, and V represent the set Aregions II, III, V, and VI represent the set B.
The common regions of A and B are II and V that represent the set\[A\cap B\].
Now, we willfind the union of the set \[A\cap B\] and set C.
Union of the regions of both the sets represents the set\[\left( A\cap B \right)\cup C\].
So, regions II, IV, V, VI, and VII represent the set\[\left( A\cap B \right)\cup C\].
(b)
First, perform the operation inside the parentheses of the set\[\left( A\cup C \right)\cap \left( B\cup C \right)\].
In the provided Venn diagram,regions I, II, IV, and V represent the set A,regions II, III, V, and VI represent the set B,andregions IV, VI, VI, and VII represent the set C.
Now, the union of regions of A and C are I, II, IV, V, VI ,and VII that represent the set\[A\cup C\].
Similarly, find set\[B\cup C\].
Union of the regions of both the sets B and C together represents the set\[B\cup C\].
Now, the common regions of the set \[A\cup C\]and set\[B\cup C\] are II, IV, V, VI, and VII. Together they represent the set\[\left( A\cup C \right)\cap \left( B\cup C \right)\].
(c)
In parts (a) and (b)set\[\left( A\cap B \right)\cup C\] and set\[\left( A\cup C \right)\cap \left( B\cup C \right)\],respectively, represent the same regions II, IV, V, VI, and VII. Both the sets are represented by same regions. So, they are equal.
Work Step by Step
.
