ECS20 Handout 9
January 27, 2011
Proof Techniques – I
1.
Direct proof
.
The implication
p
→
q
can be proved by showing that if
p
is true, then
q
must also be true. A
proof of this kind is called a
direct proof
.
Example: Prove that “If
n
is odd, then
n
2
is odd”
2.
Indirect proof
.
Since the implication
p
→
q
is equivalent to its contrapositive
¬
q
→ ¬
p
, the implication
p
→
q
can
be proved by showing that its contrapositive
¬
q
→ ¬
p
is true. This related implication is usually
proved directly. An argument of this type is called an
indirect proof
.
Example: Prove that “if 3
n
+ 2 is odd, then
n
is odd”.
3.
Proof by contradiction
.
By assuming that the hypothesis
p
is true, and that the conclusion
q
is fals, then using
p
and
¬
q
as well as other axioms, definitions, and previously derived theorems, derives a contradiction.
Proof by contradiction can be justified by noting that
(
p
→
q
)
≡
(
p
∧ ¬
q
→
r
∧ ¬
r
)
Examples:
(a) Prove that
√
2 is irrational
(b) Prove that: For all real numbers
x
and
y
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 Winter '08
 Staff
 Mathematical Induction, Natural number, Mathematical logic, Mathematical proof

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