Answer
a. $IV, V$ and $VI$
b. $IV, V$ and $VI$
c. $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$
Work Step by Step
a.
$A\cup B$ is represented by all regions in the circle for $A$ as well as all the regions in the circle for $B$: regions $I, II, III, IV, V$ and $VI$.
$C$ is represented by all regions within the circle for $C$: regions $IV, V, VI$ and $VII$
$(A\cup B)\cap C$ is represented by regions $IV, V$ and $VI$.
b.
$A\cap C$ is represented by all regions in the circle for $A$ that are also in the circle for $C$: regions $IV$ and $V$.
$B\cap C$ is represented by all regions in the circle for $B$ that are also in the circle for $C$: regions $V$ and $VI$.
$(A\cap C)\cup (B\cap C)$ is represented by regions $IV, V$ and $VI$.
c.
We can conclude that $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$
