Question

Let \[A\] and \[B\] be two sets, then find the set \[{\left( {A \cup B} \right)^\prime } \cup \left( {A' \cap B} \right)\].
A. \[A'\]
B. \[A\]
C. \[B'\]
D. None of these

Answer 1

Hint: First we will apply De Morgan’s rule in the given set. Then we will apply the reverse distribution law.

Formula used:
De Morgan’s law: The complement of the union of two sets \[A\] and \[B\] is equal to the intersection of the complement of the sets \[A\] and \[B\].
The mathematical representation is \[{\left( {A \cup B} \right)^\prime } = A' \cap B'\].
Distributive law: \[A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\].

Complete step by step solution:
Given set is \[{\left( {A \cup B} \right)^\prime } \cup \left( {A' \cap B} \right)\].
Apply De Morgan’s law in \[{\left( {A \cup B} \right)^\prime }\].
\[ = \left( {A' \cap B'} \right) \cup \left( {A' \cap B} \right)\]
Now apply reverse distribution in the above set.
\[ = A' \cap \left( {B' \cup B} \right)\]
We know that the union of a set and its complete set is a universal set.
Apply the above concept.
\[ = A' \cap U\]
The intersection of a set and a universal set is the set.
Apply the above concept.
\[ = A'\]

Hence option A is the correct option.

Additional Information:The universal set is a set that contains all elements of the sets and some more elements that do not belong to the sets.
The complement of a set A is a set which contains all elements that belong to the universal set except the elements that belong to set A.
Union of two sets: The union of two sets A and B is also a set that contains all elements of set A and set B but no element is repeated.

Note: Students often do a mistake to apply distributive property. They solve \[\left( {A' \cap B'} \right) \cup \left( {A' \cap B} \right)\] like \[A' \cup \left( {B' \cap B} \right)\]. This is incorrect way and the correct form is \[A' \cap \left( {B' \cup B} \right)\].