Question

Which is the simplified representation of \[\left( {A\prime \cap B\prime \cap C} \right) \cup \left( {B \cap C} \right) \cup \left( {A \cap C} \right)\;\]where A, B, C are subsets of set X?
A) A
B) B
C) C
D) \[X \cap (A \cup B \cup C\;)\]

Answer 1

Hint: In this question, we have to find the simplest representation of given equation of set. In order to find this use distributive law and De Morgan’s law. After that, apply the concept that intersection of any subset with universal set is equal to that subset.

Formula used: Distributive law
A. \[A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right){\rm{ }} = {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}C} \right)\]
B. \[A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right){\rm{ }} = \left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}\left( {A{\rm{ }} \cap {\rm{ }}C} \right)\]
Where A, B, C are set or subset of any universal set

De Morgan’s law
A. \[{\left( {A{\rm{ }} \cup B} \right)^c} = {A^c} \cap {\rm{ }}{B^c}\]
B. \[{\left( {A{\rm{ }} \cap B} \right)^c} = {A^c} \cup {\rm{ }}{B^c}\]
Where, \[{A^c},{B^c}\] is complement of set A and B respectively

Complete step by step solution: Given: \[\left( {A\prime \cap B\prime \cap C} \right) \cup \left( {B \cap C} \right) \cup \left( {A \cap C} \right)\;\]
By using distributive rule and De Morgan’s formula above equation becomes
\[\left( {A\prime \cap B\prime \cap C} \right) \cup \left( {B \cap C} \right) \cup \left( {A \cap C} \right)\; = \left( {(A \cup B)' \cap C} \right) \cup [\left( {A \cup B} \right)\; \cap C]\]
\[\left( {(A \cup B)' \cap C} \right) \cup [\left( {A \cup B} \right)\; \cap C] = [(A \cup B)' \cup \left( {A \cup B} \right)] \cap C\,\]
\[[(A \cup B)' \cup \left( {A \cup B} \right)] \cap C\, = X \cap C\]
\[X \cap C = C\]

Thus, Option (C) is correct.

Note: Here we must remember the algebra used in Venn diagram.
Some important properties of Sets are given below:
A. Idempotent Law is given as
(i) Union of two same sets \[A{\rm{ }} \cup {\rm{ }}A{\rm{ }} = {\rm{ }}A\]
(ii) Intersection of two same sets \[A{\rm{ }} \cap {\rm{ }}A{\rm{ }} = {\rm{ }}A\]

B. Associative Law is given as
(i) \[\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[\left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right)\]

C. Commutative Law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cup {\rm{ }}A\]
(ii) \[A{\rm{ }} \cap {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cap {\rm{ }}A\]