CS173: Discrete Mathematical Structures
Spring 2006
Homework #3
Due Sun 05/02/06, 8AM
1) Prove each problem using the indicated technique:
a) Direct Proof: Prove that the square of an even number is an even number.
b) Vacuous Proof: Show that all positive integers such that
2
logn n=
are roots of a polynomial
of the form
3 2 0ax bx cx d+ + + =
(where a, b, c, and d are fixed integers).
c) Trivial Proof: Show that all prime numbers are of the form 3k or 3k+1 or 3k-1 where k is an
integer.
d) Indirect Proof: n is even if and only if
2
5 2n+
is even (n is an integer).
e) Proof by Cases: Show that min(a, min(b, c))=min(min(a, b), c)) when a, b, and c are real
numbers.
f) Constructive Proof: For every three natural numbers x, y, z larger than 1 (x, y, z > 1), there is
some natural number larger than all x, y, z such that it is not divisible by x or y or z.
g) Non-constructive Existence Proof: Consider two numbers:
500
2 10 15!+
and
500
2 10 16!+
.
Prove that at least one of them is not a perfect square.
2) Let R, S, and T be sets. Consider the following set identity:
( ) ( ) ( )R S T R S R T=! " ! " !
a) Prove the identity using a membership table.
b) Prove the identity by showing that
( )R S T! "
is a subset of
( ) ( )R S R T! " !
, and
( ) ( )R S R T! " !
is a subset of
( )R S T! "
.
3) Let A, B, and C be sets.
a) Prove that
A B A B!=!
.
b) Prove that
( ) ( ) ( )A B C A C B C! ! =! ! !
.
c) The symmetric difference of A and B, denoted by
A B!
, is the set containing those elements in
either A or B, but not in both A and B. Show that
( ) ( )A B A B B A!=" "!
.
d) Is symmetric difference associative, that is, does it follow that
( ) ( )A B C A B C! ! =! !
?
4)
a) Let
{ ,{ }}A=! !
. What is its power set,
A
P
?
b) What is
A
P A!
?
c) If B and C are two sets, prove or disprove the identity,
B C C B!=!
.
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