Question

Let A, B, C, and D be sets. Prove that if if C ⊆ A and D ⊆ B, then C ∩ D ⊆ A ∩ B.

Answer 1

to prove X <= Y in general, you need to show that if x is a member of A then x is a member of B

Answer 2

we want to prove that if C<=D and D<= B , then (C int. D) <= (A int. B)

Answer 3

where I am using ' <= " to mean subset

Answer 4

int. = intersection

Answer 5

to prove X <= Y in general, you need to show that if x is a member of X then x is a member of Y

Answer 6

proof: Does it follow from the fact that C<=A and D<=B that (C int. D) <= (A int. B)? Thats basically what we are going to show.
So we are given C<=D and D<=B.
Next suppose that x is a member of ( C int. D).
This implies x is a member of C and x is a member of D, separately.
But this implies x is member of A, and x is a member of B.
This implies x is a member of A int. B.
Therefore we have shown that (C int. D) <= (A int. B)