Question

If $A$ is any set, then
A) \[A \cup A^{\prime} = \phi \]
B) \[A \cup A^{\prime} = U\]
C) \[A \cap A^{\prime} = U\]
D) None of the above

Answer 1

Hint: A universal set: it is denoted by $U$. It is a set that contains all the elements from all related sets, without any repetition of elements. A universal set contains a group of elements that are available in all the sets.
Union of Two Sets: Union of two given sets is the largest set which contains all the elements that are in both sets but without any repetition of elements.
A union B is given by: \[A \cup B{\text{ }} = {\text{ }}\{ x{\text{ : }}x \in A{\text{ }}or{\text{ }}x \in B{\text{ }}\} \]
The intersection of two sets: It is a set that contains all the elements that are in common to both sets A and B.
Intersection B is given by: \[A{\text{ }} \cap {\text{ }}B{\text{ }} = {\text{ }}\{ x{\text{ }}:{\text{ }}x \in A{\text{ }}and{\text{ }}x \in B\} \]
Using these definitions, we will check the given options and choose the correct answer.

Complete answer:
To solve this problem let us take an example set values such as
\[\begin{array}{*{20}{l}} {U = \left\{ {1,2,3,4,5} \right\}} \\ {A = \left\{ {1,2,3} \right\}} \\ {A^{\prime} = \{ 4,5\} } \end{array}\]
As stated in the definition of the intersection of two sets we have
\[A{\text{ }} \cap {\text{ }}A^{\prime} = \{ 1,2,3\} \cap \{ 4,5\} \]
By the definition of intersection, we have to see whether the sets have common elements or not, here there is no common element between two sets. So that we will get
$A \cap A' = \{ \} $ or $\phi $
Where $\phi $ is the empty set.
Hence option (C) is wrong
\[C)\,A \cap A^{\prime} = U\]
As stated in the definition of the union of two sets we have
\[A{\text{ }} \cup {\text{ }}A^{\prime} = \{ 1,2,3\} \cup \{ 4,5\} \]
By the definition of union, we have to combine all the elements in these sets. So that we will get
\[A{\text{ }} \cup {\text{ }}A^{\prime} = \{ 1,2,3,4,5\} \]or $U$
Where $U$ is the whole set.
Hence option (A) is wrong
\[A)\;\;\;A \cup A^{\prime} = \phi \]
Therefore from the above we have, \[\;A \cup A^{\prime} = U\] which is the option (B).
Hence, the correct option is (B)
\[B)\,A \cup A^{\prime} = U\]

Additional Information:
For example,
\[\begin{array}{*{20}{l}} {U = \left\{ {1,2,3,4,5} \right\}} \\ {A = \left\{ {1,2,3} \right\}} \\ {B = \left\{ {3,4,5} \right\}} \end{array}\]
$U$ is the universal set,
$A \cup B = \{ 1,2,3,4,5\} $ is the union of two sets,
$A \cap B = \{ \} or\phi $ is the intersection of two sets.

Note:
In the intersection of two sets both sets are the subset of set A and set B and also in the union of two sets both sets are the subset of set A and set B. The above solution can also be understood with the help of the Venn diagram.