Section 3.3
Proving set properties
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.
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.
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).
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).
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.
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
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.
Properties of set operations
Proving new properties from old
Claim. For all sets A and B,
A  (B’  A)’ = A  B
Proof.
A  (B’  A)’
= ____________ by ______
= ____________ by ______
= ____________ by ______
= ____________ by ______
= ____________ by ______
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B
by __________
Prove the following result by quoting
appropriate properties of sets.
( A  B)  ( B  C )  ( A  C )  B
Further Section 3.3 Practice
Do the Flash applets for this section as
well since they will give feedback on the
proofs.