Discrete Structures

# Lecture Notes, 11 Oct 2010

 Last Update: 11 October 2010 Note: or material is highlighted

### §§1.6: Proof Strategies (cont'd)

1. The Example from Last Time, concluded:

1. Prove: If n is odd, then n+1 is even.
Hidden assumption: n is an integer
Logical form (FOL transation):
(*)   (∀ integer n)[Odd(n) → Even(n+1)]

Strategy & Proof:

Let c be an arbitrary integer.
Show Odd(c) → Even(c+1) (and then use UG to show (*))
Suppose Odd(c) & show Even(c+1):
i.e., show ∃j[c+1=2j] (replace "Even" by its definition)
But Odd(c)
∴ ∃k[c=2k+1] (by def of Odd)
Call the k that works "k1" (by EI)
Now show (2k1+1)+1 is even:
Use algebra: (2k1+1)+1 = 2k1+2 = 2(k1+1)
So, take j = k1+1
i.e., ∃j[c+1=2j], by EG, namely: j = k1+1
c+1 is even,
i.e., Even(c+1).
Now that we know Even(c+1) on the assumption that Odd(c),
we also know Odd(c) → Even(c+1)
∴ we can infer (∀ integer n)[Odd(n) → Even(n+1)], by UG
QED

2. Here is a more formal version:

Show: (∀ integer n)[Odd(n) → Even(n+1)]
Let c be an arbitrary integer
Show Odd(c) → Even(c+1):

 1 Odd(c) :  temporary assumption to show Even(c+1) by Direct Proof 2 ∃k[c=2k+1] :  1; def of Odd 3 c=2k1+1 :  2; EI 4 c+1=2k1+1+1 :  3; algebra 5 c+1=2k1+2 :  4; algebra 6 c+1=2(k1+1) :  5; algebra 7 ∃j[c+1=2j] :  6; EG 8 Even(c+1) :  7; def of Even 9 Odd(c) → Even(c+1) :  1–8; Direct Proof 10 ∀x[Odd(x) → Even(x+1)] :  9; UG

• Only logicians and computer scientists write proofs like this,
paying attention to every detail

• mathematicians are sloppier :-)

3. Simpler ("mathematical") version:

Let c be an arbitrary integer.
Suppose Odd(c) & show Even(c+1).
Because Odd(c), we know c=2k+1, for some k.
∴ show (2k+1)+1 is even.
But (2k+1)+1 = 2(k+1)
∴ Even(c+1). QED

2. Proof by Contraposition (or: Indirect Proof):

1. Another way to prove (AB):

• try to show the logically equivalent (¬B → ¬A)

2. How? By Direct Proof:

• Suppose ¬B
& try to show ¬A

3. This is especially useful if B is "simpler" than A

4. E.g., show: If 7n–5 is odd, then n is even.
Let domain = Z

Strategy & proof:

Show (∀ int n)[Odd(7n–5) → Even(n)]

Let c be an arb. int.
& show Odd(7c–5) → Even(c):

We could supp. Odd(7c–5) & show Even(c)

• (Try it! Click on link to see what happens.)

But might be easier to show the contrapositive: (¬Even(c) → ¬Odd(7c–5)):

∴ supp. ¬Even(c), & show ¬Odd(7c–5):

But now we can use another strategy that I didn't mention:

• Get rid of negations!

i.e., instead of supposing ¬Even(c), let's supp. Odd(c),
which is mathematically equivalent (you should know that!)

and instead of trying to show ¬Odd(7c–5),
let's try to show Even(7c–5)
which is mathematically equivalent.

• Note that we are not proving anything different!
We're just restating—in a simpler form—what we're proving

So: supp. ∃k[c=2k+1]
& show ∃j[7c–5 = 2j]:

i.e., find j such that 7c–5 = 2j:

We know c=2k+1 (for some k)

∴ 7c–5 = 7(2k+1)–5
= 14k+7–5
= 14k+2
= 2(7k+1)

∴ take j = 7k+1.
QED

3. When should you use Direct Proof
and when should you use Indirect Proof (i.e., Proof by Contrapositive)?

1. In (AB), if B is simpler than A, then use Indirect Proof.

2. Else, try one strategy, and, if it goes nowhere, then try the other!

1. Another kind of indirect proof

2. To show A:
supp. ¬A & try to derive a contradiction;
i.e., try to find B such that (B ∧ ¬B)

3. Why should this work? Consider this truth-table analysis:

• (let "FALSE" name any contradiction, i.e., any proposition that is always false)

Supp. that the argument from ¬A to (B ∧ ¬ B) is valid;
∴ (¬A → (B ∧ ¬ B)) is a tautology.
but it ≡ ¬A → FALSE
∴ tval(¬A) = F (else it wouldn't be a tautology)
∴ tval(A) = T

4. E.g.: Show √2 is irrational:

• Let Q(x) = "x is rational"

• Remember: Q is the mathematical symbol for the set of rational numbers.

So we need to show: ¬Q(√2)

• Suppose, by way of contradiction, Q(√2)
& try to find B such that (B ∧ ¬B):

• Because we are assuming that Q(√2), it follows that (∃ int a,b)[(√2 = a/b) ∧ (a/b is in lowest terms)],

• i.e., a,b have no common factors

• Call this B

∴ 2 = a2/b2

∴ 2b2 = a2

∴ Even(a2)

∴ Even(a), by lemma (Rosen, p.85: 16)

• (Hint: If Even(a2), could Odd(a)?)

…to be continued…