2. Methods of Proofs

3. 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.

4. ODD AND EVEN NUMBERS Any odd integer can be expressed as Any even integer can be expressed as

5. DIVISIBILITY If a divides b, we write Example : is true because

6. RECURRENCE RELATION A recurrence relation is an equation that defines the i th value in a sequence of numbers in terms of the preceding values. Example : n! = n * (n-1)! f(n) = f(n-1) + 3 Question: Find the first 6 terms of the sequence satisfying the recurrence relation: x 1 = 3, x 2 = 2 and x n+2 = 2 x n - x n+1

7. 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”.

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

9. DIRECT PROOFS In this method of proof we prove the implication p → q by assuming that p is true and by using known 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”.

10. 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.

11. 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.

12. 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 =p ² +1, where p is a positive integer, then x is a prime number. The sum of two prime numbers is never a prime.

13. 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

14. 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 integer Q2.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 1 ² + 2 ² + … + n ² = n(n+1)(2n+1)/6 for all n ≥ 1 2 + 6 + 10 + … + (4n - 2) = 2n ² for all n > 0

15. The End