Methods of Proof

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 n2k
• ? is odd iff
• ? an integer k such that n2k1.
• 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 3x9 is even.
• Proof Suppose x is an arbitrarily chosen odd
  integer.
• Then x2k1 for some integer k. (by
  definition)
• So 3x9 3(2k1)9 (by substitution)
• 6k39 (by distributive law)
• 2(3k6) (by factoring out a 2)
  ()
• 3k6 is an integer. ()
• Hence 3x9 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 Dont 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 ndk 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 648, 55, -48, 70, 19 .

9
Properties of Divisibility

• For ?x?Z, 1x .
• For ?x?Z s.t. x?0, x0 .
• For ?a,b,c?Z, if ab and ac then a(bc) .
• Transitivity For ?a,b,c?Z,
• if ab and bc then ac .

10
Quotient-Remainder Theorem

• Theorem For ? n?Z and d?Z
• ?! q,r?Z such that
• ndqr and 0rltd.
• q is called quotient r is called remainder.
• Notation q n div d r n mod d.
• Examples 1) 53 865. 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 Thursday.
• Q. What day is Halloween this year?
• Solution There are 365 days between
• 10/31/13 and 10/31/14.
• 365 mod 7 1.
• Thus, if 10/31/13 was Thursday
• then 10/31/14 is Friday.

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 1. (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) n6k, b) n6k1, c) n6k2, d) n6k3,
• e) n6k4, f) n6k5 for some integer
  k. (3)
• n cant be 6k, 6k2 or 6k4 because
• in that case 2 n (which contradicts (1) ).
  (4)
• n cant be 6k3 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 n6k1 or n6k5.
• Lets show (2) for each of these two cases.
• Case 1 Suppose n6k1.
• Then n2 (6k1)236k212k1 (by basic
  algebra)
• 12(3k2k)1 (6)
• Let p3k2k. Then p is an integer.
• n2 12p1 . ( by substitution in (6) )
• Hence n2 mod 12 1 by quotient-remainder
  th-m.
• Case 2 Suppose n6k5. (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.