PROVE THAT A?B and C?D then A?C?B?D and AnC?BnD

Question

PROVE THAT
A⊆B and C⊆D then A∪C⊆B∪D and A∩C⊆B∩D

Answer 1

In set theory, given sets X and Y, we say that (X ⊆ Y) if and only if, for every element x ∈ X, it's also the case that x ∈ Y. X is said to be a subset of Y.

 

In general, to prove that X ⊆ Y for any sets X and Y, the strategy is usually to consider an arbitrary element x ∈ X, and then use properties of x or X and valid reasoning to prove that x must also be in Y.

 

For example, to prove that (A ∪ C) ⊆ (B ∪ D), we consider an arbitrary x in (A ∪ C), and then try to show that x must also be in (B ∪ D). If x is in (A ∪ C), then either x ∈ A or x ∈ C (or both, but we just need these two cases.) Considering these two cases:

• If x ∈ A, then x ∈ B, by definition of A ⊆ B
• If x ∈ C, then x ∈ D, by definition of C ⊆ D

 

At least one of these cases must be true, so x ∈ B or x ∈ D. And therefore x ∈ (B ∪ D), as desired.

 

You should be able to adapt this argument to the second case. Good luck!