1. Inclusion of intersection: For all sets A and B,
   (a) A Ç B Í A and (b) A Ç B Í B.
2. Inclusion in Union: For all sets A and B,
   (a) A Í A È B and (b) B Í A È B.
3. Transitive Property of Subsets: For all sets A, B, and C,
   if A Í B and B Í C, then A Í C.

The conclusion of each part of Theorem 5.2.1 states that one set is a subset of another.

Basic (Element) Method for Proving That One Set Is a Subset of Another
Let sets X and Y be given. To prove X Í Y,

1. suppose that x is a particular but arbitrarily chosen element of X,
2. show that x is an element of Y.

Examples: Prove Theorem 5.2.1 1.(b) and 2.(a)
1.(b) A Ç B Í B
Proof: Let x Î A Ç B, then x Î A Ù x Î B. In particular, x Î B. So A Ç B Í B.
2. (a) A Í A È B
Proof: Let x Î A. Now A È B = {x | x Î A Ú x Î B}. Since x Î A, we have x Î A È B. So A Í A È B.

Try it, complete this element proof.

Review Set Identities Theorem 5.2.2

Example: Prove the Absorption Law
Proof: Let sets A and B be given, then

Example 5.2.5: Prove " sets A, B, C

(A È B) - C = (A - C) È (B - C).