1. Section 3.3 Proving set properties

2. Element-wise set proofs Claim. For all sets A and B, (A  B)  A. Proof. Let sets A and B be given. Since every element of A  B is necessarily in A, this shows that (A  B)  A.

3. Element-wise set proofs Claim. For all sets A and B, (A  B)  A. Proof. Let sets A and B be given. Let x  A  B be given. This means x  A and x  B. Therefore x  A. Since every element of A  B is necessarily in A, this shows that (A  B)  A.

4. Element-wise set proofs Claim. For all sets A and B, if A  B, then A  (A  B) Proof. Let sets A and B be given such that A  B. Since every element of A is necessarily in A  B, this shows that A  (A  B).

5. Element-wise set proofs Claim. For all sets A and B, if A  B, then A  (A  B) Proof. Let sets A and B be given such that A  B. Let x  A be given. Since A  B, this implies that x  B. Since x  A and x  B, then x  A  B. Since every element of A is necessarily in A  B, this shows that A  (A  B).

6. Practice Claim. For all sets A, B, and C, if A  B and B  C, then A  C. Proof. Let sets A, B, and C be given such that ______________________________. Let x  ______ be given. _______________________________________ Therefore x  ______. Since every element of A is necessarily in C, this shows that A  C.

7. Set equality To prove that two sets A and B are equal, you must do two separate proofs: one to show that A  B and one to show that B  A. Example. We have shown that A  B  A always and that A  A  B when A  B. We can conclude from these two proofs that the following is true: If A  B, then A  B = A

8. Algebraic properties of sets Theorem. For all sets A and B, A  (A  B) = A. Proof. We must show two different things! Claim 1. A  (A  B)  A Claim 2. A  A  (A  B) This is called the absorption property and it can thought of as an algebra simplification rule. Other rules like this are given on page 215.

9. Properties of set operations

10. Proving new properties from old Claim. For all sets A and B, A  (B’  A)’ = A  B Proof. A  (B’  A)’ = ____________ by ______ = ____________ by ______

11. Proving new properties from old Claim. For all sets A and B, A  (A’  B) = A  B Proof. A  (A’  B) = (A  A’)  (A  B) by _______ = U  (A  B) by __________ = (A  B)  U by __________ = A  Bby __________

12. Prove the following result by quoting appropriate properties of sets.

13. Further Section 3.3 Practice Do the Flash applets for this section as well since they will give feedback on the proofs.