Question

If the cardinality of A is 3, the cardinality of B is 5, A and B are not disjoint and A is a not subset of B, determine the lower and upper bound for the cardinality of A ∩ B .

Answer 1

#(A) = 3, #(B) = 5, A & B are not disjoint, so, #(A intersection B) > 0

And, A is not subset of B,

So, (A intersection B) is subset of both A and B, hence contained in both of them.

If A was the subset of B, then, (A intersection B) would be A and would have cardinality 3, which could have been the maximum possible cardinality of (A intersection B)

But, since this is not the case, so we have, the maximum cardinality of (A intersection B) to be less than 3, which is 2.

And since A and B are disjoint, so (A intersection B) will have cardinality more than 0, which is 1.

So, we got,

1 <= #(A intersection B) <= 2

Upper bound is 2 and lower bound is 1