Answer
Set \[\left( A\cup B \right)'\]represented by regions IV.
Set \[A'\cap B'\]represented by regions IV. Both the sets are represented by same region, so they are equal.
Work Step by Step
First, perform the operation inside the parenthesis of the set\[\left( A\cup B \right)'\].
So we compute\[A\cup B\].
Set \[A\cup B\] contains all the elements which are either in set A or set B or in both.
In the Venn diagram,
Regions II, III represent the set B.
Regions I, II represent the set A.
Now the union of regions of set A and set B are I, II, III. So it represents the set\[A\cup B\].
To find the complement of the set\[A\cup B\], it contains all the elements of the universal set Uexcept the elements of set\[A\cup B\].
So region IV represents the set\[\left( A\cup B \right)'\].
Similarly find the complement of the set A and set B.
In the Regions III and IV represent the set\[A'\].
And Regions I and IV represent the set\[B'\].
Then common regions of both the sets\[A'\]and\[B'\]is region IV only. Hence region IV represents the set \[A'\cap B'\]
Therefore,
Set \[\left( A\cup B \right)'\]represented by regions IV.
Set \[A'\cap B'\]represented by regions IV.
Both the sets are represented by same region, so they are equal.
Hence,
\[\left( A\cup B \right)'=A'\cap B'\]
