Answer: Hello!
ok, remember that "if and only if" implies that you need to prove the statement in both ways, this is represented with the ⇔ usually.
a) For any sets A, B and C, A ∩ B ⊆ C if and only if either A ⊆ C or B ⊆ C.
In this type of problems, i find very useful start looking for some counterexample.
In this case, suppose that A = {1,2,3,4,5} , B = {3,4,5,6,7} and C = {3,4,5,6}
then is easy to see that A ⊄ C and B ⊄C.
And A∩B = {3,4,5}
then A∩B ⊂ C
then the statement is false (because one of the ways is false, remember that this is an "if and only if" statement)
b) For any sets A, B and C, A ⊆ B ∩ C if and only if both A ⊆ B and A ⊆ C.
the first way is true; because if A ⊆ B ∩ C. then all the elements of A are in the intersection of B and C (which are common elements for B and C) and then all the elements of A are in the set B and in the set C, and this means that A ⊆ B and A ⊆ C.
But let's see the other way now, suppose that A ⊆ B and A ⊆ C, now we want to know if A ⊆ B ∩ C.
if A ⊆ B and A ⊆ C, means that all the elements of A are in B, and all the elements of A are in C, then all the elements of A are common elements between B and C, this means that B ∩ C is at least equal to A (at least, because we know that all the elements of A are common elements between B and C, but there could be more common elements that don belong to A)
then A ⊆ B ∩ C.
