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Introduction
Theorems and Proofs
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Theorems and Proofs
102
Previous Lecture
Axioms and theorems
Rules of inference for quantified statements
Direct proofs
Discrete Mathematics – Theorems and Proofs
103
Methods of Proving – Proof by Contraposition
Sometimes direct proofs do not work
Prove that if
3
n
+ 2
is even, then
n
is also even
That is
2200
x
(E(
3
x
+ 2
)
→
E(
x
))
Let us try the direct approach:
As for the generic value
n
the number
3
n
+ 2
is even, for some
k
we have
3
n
+ 2 = 2
k
.
Therefore
3
n
= 2(
k
+ 1).
Now what?
What if instead of
2200
x
(E(
3
x
+ 2
)
→
E(
x
))
we prove the
contrapositive,
2200
x
(
¬
E(
x
)
→ ¬
E(
3
x
+ 2
)) ?
Definition:
n
is
even
if and only if there is
k
such that
n =
2
k
Discrete Mathematics – Theorems and Proofs
104
Methods of Proving – Proof by Contraposition
(cntd)
So assume that
n
is
odd
,
that is there is
k
such that
n
= 2
k
+ 1
.
Then
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 Spring '08
 PEARCE

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