Answer
Not true
Work Step by Step
First, perform the operation inside the parentheses of the set\[A\cup \left( B\cap C \right)\].
Now, compute\[B\cap C\].
Set \[B\cap C\] contains all the common elements of set B and set C.
In the provided Venn diagram,regions IV, V, VI, and VII represent the set C,regions II, III, V, and VI represent the set B,andregions I, II, IV, and V represent the set A.
The common regions of set B and set C are V and VI, so it represents the set\[B\cap C\].
Now, find the union of the set \[B\cap C\] and set A.
Take union of regions of both the sets that together represent the set\[A\cup \left( B\cap C \right)\].
So, regions I, II, IV, V, and VI represent the set\[A\cup \left( B\cap C \right)\].
Perform the operation inside the parentheses of the set\[\left( A\cup B \right)\cap C\].
Now, compute\[A\cup B\].
Set \[A\cup B\] contains all the elements thatare in the set A,set B, or in both.
In the provided Venn diagram, take union of the regions of set A and set B asI, II, III, IV, V, VI,and V, which together represent the set\[A\cup B\].
Now, find the intersection of the set \[A\cup B\] and set C.
In the Venn diagram, common regions of both sets are IV, V, and VI, which together represent the set\[\left( A\cup B \right)\cap C\].
Therefore, set \[A\cup \left( B\cap C \right)\]is represented by regions I, II, IV, V, and VI and set \[\left( A\cup B \right)\cap C\]is represented by regions IV, V, and VI.
Both the sets are represented by different regions, so they are not equal.
