Question

If $A$ and $B$ are any 2 sets and $A$, $B$ are subsets of $U$ then $A \cup A'$ is
(a) $U$
(b) $\phi $
(c) $A$
(d) $A'$

Answer 1

Hint: We can do this question by drawing a Venn-diagram of $A \cup A'$. We will let the sets randomly as $A = \left\{ {1,2} \right\}$, $B = \left\{ {2,3} \right\}$ and $U = \left\{ {1,2,3,4} \right\}$ such that it satisfies the given condition. Take the compliment of $A$ and then take its union of $A'$. Write the value of the set $A \cup A'$.

Complete step by step Answer:

We are given that the sets $A$ and $B$of the $U$.
The universal set is the set which is the bigger set of all the sets.
Let $U$ be the set such that includes both sets $A$ and $B$.
Represent the given information using a Venn diagram.
Let $A = \left\{ {1,2} \right\}$, $B = \left\{ {2,3} \right\}$ and $U = \left\{ {1,2,3,4} \right\}$ which satisfies the condition that $A$ and $B$are subsets of $U$.
Then, we will have

We want to find the value of the $A'$, which is the complement of $A$.
Compliment of the set includes all the elements of the universal set except the elements of the set.
Then, $A' = \left\{ {2,3,4} \right\}$
We have to find the value of $A \cup A'$
Which implies $\left\{ {1,2} \right\} \cup \left\{ {2,3,4} \right\} = \left\{ {1,2,3,4} \right\}$
Which is also equal to $U$

Hence, $A \cup A' = U$
Hence, option A is correct.

Note: It is the property of sets that union of a set with its complement is the universal set as it will include all the elements of the universal set. When we take the union of the two sets then we take all the elements of both the sets.