1. Methods of Proof Proof techniques in this handout
Direct proof
Division into cases
Proof by contradiction
In this handout, the proof techniques will be used to prove properties in number theory.

2. Even and Odd Integers Definition: An integer n ● is even iff
 an integer k such that n=2k;
● is odd iff
 an integer k such that n=2k+1.
Ex: If x and y are integers,
is even or odd?

3. Method of Direct Proof To prove a statement: xD if P(x) then Q(x).
Suppose x is a particular but arbitrarily chosen element of D
for which P(x) is true;
Show the conclusion Q(x) is true by using
♦ definitions;
♦ previously established results;
♦ rules of logical inference.

4. Method of Direct Proof (Ex.)
Show xZ if x is odd
then 3x+9 is even.
Proof: Suppose x is an arbitrarily chosen odd integer.
Then x=2k+1 for some integer k. (by definition)
So 3x+9 = 3(2k+1)+9 (by substitution)
= 6k (by distributive law)
= 2(3k+6) (by factoring out a 2) (*)
3k+6 is an integer (**)
Hence 3x+9 is even
based on (*), (**) and definition of even integers ▀
(this is what we needed to show)

5. Directions for writing proofs
Write the theorem to be proved.
Clearly mark the beginning of your proof
with the word Proof.
3) Make your proof self-contained.
(Identify all variables used in the proof;
state the sources of outside facts).
4) Write proofs in complete English sentences.

6. Common mistakes in proofs
Arguing from examples
Using same letter
to mean two different things
Jumping to a conclusion
(without adequate reasons)

7. Types of Mathematical Statements
Theorems: Very important statements that
have many and varied consequences.
Propositions: Less important and consequential.
Corollaries: The truth can be deduced almost immediately
from other statements.
Lemmas: Don’t have much intrinsic interest but help to prove other theorems.

8. Divisibility
Definition: For n,d Z and d≠0 we say that n is divisible by d
iff n=d·k for some k Z .
Alternative ways to say:
n is a multiple of d , d is a factor of n ,
d is a divisor of n , d divides n .
Notation: d | n .
Examples: 6|48, 5|5, -4|8, 7|0, 1|9 .

9. Properties of Divisibility
For xZ, 1|x .
For xZ s.t. x≠0, x|0 .
For a,b,cZ, if a|b and a|c then a|(b+c) .
Transitivity: For a,b,cZ,
if a|b and b|c then a|c .

10. Quotient-Remainder Theorem
Theorem: For  nZ and dZ+
! q,rZ such that
n=d·q+r and 0≤r<d.
q is called quotient; r is called remainder.
Notation: q = n div d; r = n mod d.
Examples: 1) 53 = 8·6+5. Hence
53 div 8 = 6; 53 mod 8 = 5.
2) -29 = 7·(-5)+6. Hence
-29 div 7 = -5; -29 mod 7 = 6.

11. Example of using div and mod
Last year Halloween was on Monday.
Q.: What day is Halloween this year?
Solution: There are 365 days between
10/31/16 and 10/31/17.
365 mod 7 = 1.
Thus, if 10/31/16 was Monday
then 10/31/17 is Tuesday.

12. Proof Technique: Division into Cases
Suppose at some stage of a proof
● we know that
A1 or A2 or A3 or … or An is true;
● want to deduce a conclusion C.
Use division into cases:
Show A1→C, A2→C, …, An→C.
Conclude that C is true.

13. Division into Cases: Example
Proposition: If nZ such that
neither of 2 or 3 divide n, (1)
then n2 mod 12 = (2)
Proof: Suppose nZ s.t. neither of 2 or 3 divide n.
By quotient-remainder theorem,
exactly one of the following is true:
a) n=6k, b) n=6k+1, c) n=6k+2, d) n=6k+3,
e) n=6k+4, f) n=6k+5 for some integer k (3)
n can’t be 6k, 6k+2 or 6k+4 because
in that case 2 | n (which contradicts (1) ). (4)
n can’t be 6k+3 because in that case 3 | n
(which contradicts (1) ) (5)

14. Division into Cases: Example(cont.)
Proof(cont.): Based on (3), (4) and (5),
either n=6k+1 or n=6k+5.
Let’s show (2) for each of these two cases.
Case 1: Suppose n=6k+1.
Then n2 = (6k+1)2=36k2+12k+1 (by basic algebra)
= 12(3k2+k) (6)
Let p=3k2+k. Then p is an integer.
n2 = 12p+1 . ( by substitution in (6) )
Hence n2 mod 12 = 1 by quotient-remainder th-m.
Case 2: Suppose n=6k+5. (exercise) ■

15. Method of Proof by Contradiction
Suppose the statement to be proved is false.
Show that this supposition logically leads to a contradiction.
Conclude that the statement to be proved is true.
Example of proof by contradiction.
Theorem: There is no greatest integer.
The proof on the board.
We will see several contradiction proofs in graph theory.