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Discussion #12
Chapter 1, Sections 1.6.6
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Discussion #12
Deduction, Proofs
and Proof Techniques
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Chapter 1, Sections 1.6.6
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Topics
•
Proofs = Sound Arguments, Derivations,
Deduction
•
Proof Techniques
1.
Exhaustive
2.
Equivalence to Truth
3.
If and only if (iff)
4.
Contrapositive Proof
5.
Contradiction (or Indirect)
6.
Conditional
7.
Case Analysis
8.
Induction
9.
Direct
Discussion #12
Chapter 1, Sections 1.6.6
3/21
1. Exhaustive Truth Proof
Show true for all cases.
–
e.g.
Prove x
2
+ 5 < 20 for integers 0
≤
x
≤
3
0
2
+ 5 = 5
< 20
T
1
2
+ 5 = 6
< 20
T
2
2
+ 5 = 9
< 20
T
3
2
+ 5 = 14
< 20
T
–
Thus, all cases are exhausted and true.
–
Same as using truth tables
when all
combinations yield a tautology.
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2. Equivalence to Truth
2a.
Transform logical expression to T.
Prove:
R
∧
S
⇒
¬
(
¬
R
∨
¬
S)
R
∧
S
⇒
¬
(
¬
R
∨
¬
S)
≡
R
∧
S
⇒
¬¬
R
∧
¬¬
S
de Morgan’s law
≡
R
∧
S
⇒
R
∧
S
double negation
≡
¬
(R
∧
S)
∨
(R
∧
S)
implication (P
⇒
Q
≡
¬
P
∨
Q)
≡
T
law of excl. middle (P
∨
¬
P)
≡
T
Discussion #12
Chapter 1, Sections 1.6.6
5/21
2. Equivalence to Truth (continued…)
2b.
Or, transform lhs to rhs or vice versa.
Prove:
P
∧
Q
∨
¬
P
∧
Q
⇔
Q
P
∧
Q
∨
¬
P
∧
Q
≡
(P
∨
¬
P)
∧
Q
distributive law (factoring)
≡
T
∧
Q
law of excluded middle
≡
Q
identity
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3. If and only if Proof
•
Also called necessary and sufficient
•
If we have P
⇔
Q, then we can create two
standard deductive proofs, namely,
(P
⇒
Q) and (Q
⇒
P).
•
i.e. P
⇔
Q
≡
(P
⇒
Q)
∧
(Q
⇒
P)
•
Transforms proof to a standard deductive
proof
actually two of them
Discussion #12
Chapter 1, Sections 1.6.6
7/21
3. If and only if Proof: Example
Prove: 2x – 4 > 0 iff x > 2.
Thus, we can do two proofs:
(1) If 2x – 4 > 0 then x > 2.
(2) If x > 2 then 2x – 4 > 0.
Proof:
(1) Suppose 2x – 4 > 0.
Then 2x > 4 and thus x > 2.
(2) Suppose x > 2.
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This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.
 Winter '12
 MichaelGoodrich

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