Question

If A = {1,2,3,4} and B = {2,4,6} find the no of sets C such that A intersection B is a subset of C and A U B is a superset of C

Answer 1

Answer:

We'll start with your problem as a motivation to prove the general case.

We've been given A={1,2,3,4}A={1,2,3,4} and B={2,4,6}B={2,4,6} and we need to find the number of sets CC such that A∩B⊆C⊆A∪BA∩B⊆C⊆A∪B and in particular {2,4}⊆C⊆{1,2,3,4,6}{2,4}⊆C⊆{1,2,3,4,6}. Since, {2,4}{2,4} is a subset of CC, 22 and 44 must be in CC.

Now, consider the set (A∪B)∖(A∩B)(A∪B)∖(A∩B) and call it DD. Thus, in this case D=(A∪B)∖(A∩B)={1,3,6}D=(A∪B)∖(A∩B)={1,3,6}. Now, let's construct the power set of DD and let's call it PDPD. Pick any arbitrary element F∈PDF∈PD (which is nothing but a set disjoint to A∩BA∩B) and define C:={2,4}∪FC:={2,4}∪F. Notice that this set CC satisfies the necessary condition i.e. A∩B⊆C⊆A∪BA∩B⊆C⊆A∪B. Since DD contains 33 elements, hence PD

Step-by-step explanation:

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Answer 2

A = {1, 2, 3, 4}

B = {2, 4, 6}

A∩B = {2, 4}

There can be infinitely many sets C such that A∩B is a subset of C

For example, C = {2,4,6} or C = {2,4,6,8} or C is a set of all even natural numbers, the list is infinite!

A∪B = {1, 2, 3, 4, 6}

We have to find number of set C such that A∪B is a superset of C, or C is a subset of A∪B

Number of elements in A∪B = 5

So, number of sets C possible such that A∪B is a superset of C = 2⁵ = 32