Question

Which one of the following is correct?
A) \[A \cup (B - C) = A \cap (B \cap C\prime )\]
B) \[A - (B \cup C) = (A \cap B\prime ) \cap C\prime \]
C) \[A - (B \cap C) = (A \cap B\prime ) \cap C\]
D) \[A \cap (B - C) = (A \cap B) \cap C\]

Answer 1

Hint: In this question, we have to find which of the given condition is correct. In order to find this concept of Venn diagram is used. Apply concept of Venn diagram on given options to get the correct option.

Formula used: In this question we are going to use the Venn diagram. This diagram give the relation between various set and their subset.

Complete step by step solution: Draw a Venn-diagram taking three intersecting sets A, B and C under a universal set X. After intersection eight regions will be developed.
Region a, b, d, e are lies in set A
Region b, c, e, f are lies in set B
Region d, e, f, g are lies in set C
Region a, b, c, h are lies in set C’
Region a, b, g, h are lies in set B’
Now we have first option \[A \cup (B - C) = A \cap (B \cap C\prime )\]
\[LHS = (a,b,d,e) \cup (b,c,e,f - d,e,f,g)\]
\[LHS = (a,b,d,e) \cup (b,c,d,g) = (a,b,c,d,e,g)\]
\[RHS = (a,b,d,e) \cap (b,c,e,f \cap a,b,c,h)\]
\[RHS = (a,b,d,e) \cap (b,c) = b\]
LHS is not equal to RHS
Now we have second option \[A - (B \cup C) = (A \cap B\prime ) \cap C\prime \]




\[RHS = (a,b,d,e \cap a,d,g,h) \cap (a,b,c,h) = a\]
\[LHS = (a,b,d,e) - (b,c,d,e,f,g) = a\]
Hence option \[A - (B \cup C) = (A \cap B\prime ) \cap C\prime \]is correct



Thus, Option (B) 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) \[A{\rm{ }} \cap {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cap {\rm{ }}A\]

D. Distributive law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right){\rm{ }} = {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right){\rm{ }} = \left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}\left( {A{\rm{ }} \cap {\rm{ }}C} \right)\]
Where A, B, C are set or subset of any universal set

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