Answer:
The set that is equivalent to (C∩ Aˉ ) U (C∩B) is (C ∩ Aˉ ) U B. Option (c) is true.
Explanation:
Distributivity of Union: Let's first apply the distributive property of union over intersection:
(C∩ Aˉ )∪(C∩B) = C∩(Aˉ ∪ B)
Commutativity of Intersection: Now, we can switch the order of the intersections:
C∩(Aˉ ∪ B) = C∩(B ∪ Aˉ )
Idempotence of Intersection: Since intersecting with the same set twice doesn't change the result, we have:
C∩(B ∪ Aˉ ) = C∩(Aˉ ∪ B)
Therefore, all the options except (C∩ Aˉ )∪B are incorrect because they represent different sets.
Here's a breakdown of the other options:
a. C∩( Aˉ ∪B): This option is incorrect because it only considers the elements that are in C and also in the union of Aˉ and B. It does not consider the elements that are in C and only in B.
b. C∪( Aˉ ∩B): This option is incorrect because it considers the elements that are in C or in the intersection of Aˉ and B. It does not consider the elements that are in both C and B.
c. (C∪ Aˉ )∩B: This option is incorrect because it considers the elements that are in the union of C and Aˉ and also in B. It does not consider the elements that are only in C and B.
Thus, the correct answer is Option (c).