Meaning of Proof Methods of Proof

Transcription

1 Mathematical Proof Meaning of Proof Methods of Proof 1 Dr. Priya Mathew SJCE Mysore Mathematics Education 4/7/2016

2 2 Introduction Proposition: Proposition or a Statement is a grammatically correct declarative sentence which makes sense, which is either true or false, but not both. Example: Sum of 2 and 5 is 7 ( proposition which is true) Sum of 2 and 6 is 7 - (Proposition which is false) Is 7 the sum of 2 and 5? -(not a proposition, it is not a declarative sentence ) Sum of 2 and 5 is green. ( not a proposition as the sentence does not make any sense) Majula is a good student ( sentence is vague and does not have a definite truth value) The method of establishing the truth of a proposition is known as proof Dr. Priya Mathew SJCE Mysore Mathematics Education 4/7/2016

3 Method of establishing the logical validity of the conclusion of a theorem, as a consequence of the premise, axioms, definitions and already established theorems of the mathematical system is known proof. Proof is a deductive argument for a mathematical statement rather than inductive or empirical argument. In the argument previously established statements or theorems can be used A process which can establish the truth of a mathematical statement based purely on logical arguments 3

4 The word comes from the Latin Word Probare means to test Probe (English) : to check in detail Probar (Spanish) : to taste, smell, to touch or to test Provare (Italian) : to try Probieren (German) : to try 4

5 A statement that is proved is often called as a theorem. Once a theorem is proved it can be used as the basis to prove further statements. A proof must demonstrate that a statement is always true, rather than enumerating many conformity cases. 5

6 Verify that : 1. The product of two even number is even 2. Sum of any even number is even 3. Sum of the interior angles of a triangle is 180 In verification, we cannot physically check the products of all possible pairs of even numbers It may help us to make statement we believe is true. We cannot be sure that it is true for all cases. Verification can often be misleading 6

7 To establish that a mathematical statement is false, it is enough to produce a single counter example 7+5 = 12 is a counter example to the statement that the sum of two odd numbers is odd. there is no need to establish the validity of a mathematical statement by checking or verifying it for thousands of cases. 7

8 a) Direct Proof b) Indirect Proof c) Proof by counter examples d) Proof by induction 8

9 It is the most familiar proof It is to prove statements of the form if P then Q. i.e P Q. This method of proof is to take an original statement P, which we assume to be true and use it to show directly that another statement Q is true. Example : If a, b are two odd natural numbers, then the product of a and b is also an odd number 9

10 Proof: Any odd number can be written as 2n+1, where n is any natural number. Let a = 2n+1 and b = 2m+1 So a.b = (2n+1) (2m+1) = 4mn + 2n+ 2m+1 = 2 (2mn + m +n) +1 = 2(p +1) where, p = 2mn + m +n Therefore, 2p+1 is an odd number. a.b is an odd number Eg. 2: Prove that product of two even natural numbers is even 10

11 It is synonyms with proof by contradiction In this, a statement to be proved is assumed false for the sake of reasoning, and if the assumption leads to an impossibility or contradiction, then the statement assumed false in the beginning is proved to be true. The idea behind it is that if what you assumed creates a contradiction, the opposite of your initial assumption is the truth. 11

12 1. Proof by contradiction: It is used normally, when it is not straight forward as to how to proceed with the proof. We assume to the contrary that the conclusion is false and by logical arguments arrive at something which is absurd. 12

13 Prove that sum of any even number is even Proof: Let us assume n is an odd number, where n> n is an odd number. 2[n(n+1)/2] is an odd number n(n+1) is an odd number Which is a contradiction, since n(n+1) is always an even number. Hence the proof. 13

14 Assume that the conclusion is false Establish a contradiction : Establish that the assumption which was considered as false contradicts some previously existing theorem, definition, postulates, etc. State that the assumption must be false, thus the conclusion or the statement to be proved is true. 14

15 2. Proof by contra positive : It is based on the fact that a proposition p q is equivalent to its contra positive q p We assume that the negation of the conclusion is true and in straight forward way show that the negation of the hypothesis is true. i.e if the conclusion is false, then the premises are false. But the premises are true, therefore, the conclusion must be true. 15

16 Theorem: If n 2 is odd then n is odd Proof : its contrapositive is if n is not odd then n 2 is not odd To prove contrapositive assume that n is not odd Therefore n = m for some integer m therefore n 2 = (2m) 2 = 4 m 2 = 2(2 m 2 ) is even Hence n 2 is not odd Therefore, If n 2 is odd then n is odd 16

17 If we have a statement involving a universal quantifier and to prove that the statement is true, we have to show that the statement is true for every element in the universal set. To show that such a statement is false, it is sufficient to show that there is a particular value in the universal set for which the statement is false. Exhibiting such value for which the statement is false is known as counter example. Method of giving such a counter example to show that such a statement is false is known as method of disproof or proof by counter example 17

18 Statement : Every odd natural number is a prime To disprove this, it is sufficient to give example of one odd natural number which is not a prime. Consider a natural number 9 it is odd as 9 = 2x4 +1 however it is not a prime since 9 = 3x3 So, 9 is an odd natural number which is not a prime. Hence, the given statement is not true. 18

19 The sum of two odd numbers is odd Counter Example, 7+5 =12 19

20 Mathematical induction is a mathematical proof technique used to establish a given statement for all natural numbers, although it can be used to prove statements about any well-ordered set. It is a form of direct proof, and it is done in two steps. 20

21 The basis (base case): prove that the statement holds for the first natural number n. Usually, n = 0 or n = 1, rarely, n = 1 The inductive step: is to prove that the given statement for any one natural number implies the given statement for the next natural number. if the statement holds for some natural number n, then the statement holds for n

22 From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers. 22

23 Example Prove that n = n(n + 1) / 2, for any integer n 1. Dr. Priya Mathew SJCE Mysore 4/7/ Mathematics Education

24 1. To prove a theorem, we should have a rough idea as how to proceed 2. The information already given to us in a theorem (i.e. the hypothesis) has to be clearly understood and used. 3. A proof is made up of a successive sequence of mathematical statement. Each statement in a proof is logically deduced from a previous statement in the proof, or from a theorem proved earlier or an axiom, or our hypothesis. 4. The conclusion of a sequence of mathematically true statements laid out in a logically correct order should be what we want to prove, that is, what a 4/7/2016 theorem claims. Dr. Priya Mathew SJCE Mysore Mathematics Education 24