Methods of Proofs

PREDICATE LOGIC • The “Quantifiers” and are known as predicate

quantifiers. " means for all and means there exists. • Example 1: If we have one computer that all students must

share, we say: one computer students.

• Example 2: If each student has a separate computer, we would say: students one computer.

• Note: The two example may both appear to be saying that “there exists” only one computer; however, example 2 is actually saying mathematically that every student has a ‘unique’ computer.

ODD AND EVEN NUMBERS • Any odd integer can be expressed as

• Any even integer can be expressed as

12 pn

pn 2

DIVISIBILITY • If a divides b, we write

• Example : is true because

ba |

18|3

6318

RECURRENCE RELATION • A recurrence relation is an equation that defines

the ith value in a sequence of numbers in terms of the preceding values.

Example : n! = n * (n-1)! f(n) = f(n-1) + 3Question: Find the first 6 terms of the sequence

satisfying the recurrence relation:

x1 = 3, x 2 = 2 and xn+2 = 2 x n - x n+1

1i

LOGIC • The inverse of is • The converse of is q → p• The contra positive of is Example : Write the inverse ,converse and contra

positive of the statement ,“If John is intelligent then he will pass the exam”.

qp qp ~~

qp

qp pq ~~

Methods of Proofs • Direct proofs • Indirect proofs or Contrapositive proofs• Proof by contradiction• Proof by counter Example• Proof by Mathematical Induction

DIRECT PROOFS In this method of proof we prove the implicationp → q by assuming that p is true and by usingknown facts, rules and theorems we try to show

that q is also true. Example : Prove by using direct proofs “ If x is even then x² is even”.

CONTRAPOSITIVE PROOFS OR INDIRECT PROOFS

In this method of proof we try to prove the implication p → q by assuming that q is false and by using known facts ,rules and theorems we try to show that p is also false .

Example : Prove if n² is even then n is also even by contrapositive.

PROOFS BY CONTRADICTION • A proof by contradiction is a proof of an

implication that shows that joining the assumption “Q is false” together with the premise “P is true” leads to a contradiction. In other words in this method of proof we prove the implication p → q by assuming that p ᴧ q ̴� is true and by using known facts ,rules and theorems we try to show that our assumption is false .

• Example: If n and m are odd integers, then n + m is an even integer .Give a proof by contradiction.

COUNTEREXAMPLES • At times use of proofs is not only impossible, but

unnecessary. Sometimes in order to prove something all that is necessary is to provide an example that proves the statement false, i.e. a counterexample.

Example:Find a counterexample to show that the following are

false• If x =p² +1, where p is a positive integer, then x is a

prime number.• The sum of two prime numbers is never a prime.

MATHEMATICAL INDUCTION• In order to prove something by induction you need to first

prove the base step i.e. for n=0,1….. (Basis Step )

• The next step is to assume that the proof is true for some value k. (Inductive Hypothesis)

• Then you must prove that it is also true for k + 1(Inductive Step)

• After proving it for k + 1 you have proved it true by induction.

• Example: Prove by Induction, 1+2+3+……..+n = n(n+1)/2

Exercises …..Q1.Give a direct proof for each of the following:• If n and m are even integers, then n + m is an even integer• If n and m are even integers, then n - m is an even integerQ2.Give a proof by contradiction, If n is an even integer and m is an odd integer, then n + m

is an odd integer.Q3. Prove the followings by Mathematical Induction• 1² + 2² + … + n² = n(n+1)(2n+1)/6 for all n ≥ 1 • 2 + 6 + 10 + … + (4n - 2) = 2n² for all n > 0

The End