1
Handout #22
CS103
April 18, 2011
Robert Plummer
Problem Set #3 Solutions
1.
Suppose A and B are sets.
Prove that (A
B) - A = B - (A
B).
PROOF:
We will prove equality by showing that LHS
RHS and RHS
LHS.
[We'll put some whitespace in the first half of the proof to show how it improves readability.]
Suppose x
((A
B) – A).
Then (x
A
x
B)
x
A by the definition of union and set difference.
Since x
A and x
A cannot both be true, we have x
B
x
A.
If x
A, then x
A
B, so we have x
B
x
(A
B), which means x
(B – (A
B)) by the definition
of set difference.
So LHS
RHS.
For the second half of the proof, suppose x
(B – (A
B)).
Then x
B
x
(A
B) by the definition of set
difference.
Thus x
B
(x
A
x
B) by definition of intersection, and x
B
x
A since x
B and
x
B cannot both be true.
Since x
B, x
A
B, so we have x
(A
B)
x
A and x
((A
B) – A) )
by the definition of set difference.
So RHS
LHS.
■
2.
Consider the following three conditions:
(i)
A
B
(ii)
A
C
(iii)
A
B - C =
Can there exist sets A, B, and C that satisfy all three conditions?
If so, provide an
example.
If not, provide a proof to that effect.
Example:
A = {1, 2}, B = {2, 3}, C = {2, 4}.
Or, any non-empty sets where A = B = C.
3.
If A and B are sets, is it possible that A
B and A
B?
Give an example or prove that
this is not possible.
Example: A = {1, 2}, B = {1, 2, {1, 2}}
4.
Suppose that A and B are sets.
Prove that B
A = A if and only if A
B.
[Indenting can improve readability, and placing a formula on a separate line makes it convenient to give it a
number for later reference.
Lower case Roman numerals are often used for this purpose.]
PROOF.
First the forward direction (
):
Suppose that (B
A) = A
(i)
and suppose that x
A.
Then by (i), x
B
A., so x
B.
Thus A
B.
Now the backward direction (
):
Suppose that A
B
(ii)
and suppose
that x
B
A.
Then x
A.
Now suppose x
A.
Then x
B by (ii), and x
B
A.
Thus B
A = A.
■

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