CS103
HO#9
More proofs, SAT Solving
4/6/11
1
CS103
Mathematical
Foundations
of Computing
4/6/11
Monday
Kevin
10 - 12
Gates 200
Bob
3:45 - 5:00
Gates 178
Tuesday
Ryan
1:15 - 3:15
???
Hrysoula
7 - 9
Gates B12
Wednesday
Conal
12:15 - 2:15
Gates 200
Bob
3:45 - 5:30
Gates 178
Thursday
Evan
2:15 - 4:15
Gates 400
Mingyu
7 - 9
200-201
Sunday
Neel
1 - 3
Gates B12
Karl
5 - 7
Gates B12
Office Hours
Proof by Contradiction
Indirect Proof
Suppose ¬Q
...
...
¬S
Contradiction: S and ¬S
Q
S
...
...
Trying to prove Q
S is a premise or a
derived statement
Suppose
n
is prime and
n
> 2.
Prove that
n
is odd.
PROOF by contradiction
:
Suppose that
n
is even.
Then there is an integer
k
such that
n
= 2
k
, and
since
n
> 2,
k
> 1.
But that means that
n
is divisible by 2 and
k
, and thus
n
is not prime, contradicting the hypothesis.
Thus
n
must be odd.
■
n
is prime
n
> 2
Suppose
n
is even
Then
n
= 2
k
for some integer
k
3, def. of even
k
> 1
2, 4 algebra
n
is divisible by 2 and
k
4, def. of divides
n
is not prime
5, 6 def. of prime
Contradiction
1, 7
n
is odd
3 - 8
1
2
3
4
5
6
7
8
9
So
¬
Q follows from P
1
, …, P
n
Proof by Contradiction
Indirect Proof
Suppose that
from premises P
1
, …, P
n
we wish to show
¬
Q
.
One possibility is to temporarily
assume Q
, and see if
we can
show that P
1
, …, P
n
, Q is impossible
, i.e., that
these claims can’t be true simultaneously.
Then in any circumstances where P
1
, …, P
n
are true,
Q is false
¬
Q is true

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