mathematical proof - mathematical induction is a mathematical proof technique used to establish a...
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Meaning of Proof
Methods of Proof
4/7/2016 1 Dr. Priya Mathew SJCE Mysore Mathematics Education
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
4/7/2016 2 Dr. Priya Mathew SJCE Mysore Mathematics Education
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
4/7/2016 3 Dr. Priya Mathew SJCE Mysore Mathematics Education
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
Provare (Italian) : to try
Probieren (German) : to try
4/7/2016 4 Dr. Priya Mathew SJCE Mysore Mathematics Education
A statement that is proved is often called as a
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
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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
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To establish that a mathematical statement is
false, it is enough to produce a single
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.
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a) Direct Proof
b) Indirect Proof
c) Proof by counter examples
d) Proof by induction
4/7/2016 8 Dr. Priya Mathew SJCE Mysore Mathematics Education
It is the most familiar proof It is to prove statements of the form if P then
Q . i.e PQ. 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
4/7/2016 9 Dr. Priya Mathew SJCE Mysore Mathematics Education
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 4/7/2016 10 Dr. Priya Mathew SJCE Mysore Mathematics Education
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.
4/7/2016 11 Dr. Priya Mathew SJCE Mysore Mathematics Education
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.
4/7/2016 12 Dr. Priya Mathew SJCE Mysore Mathematics Education
Prove that sum of any even number is even Proof: Let us assume + +6+8+…….+ n is an odd
number, where n>0 + + + +…….+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.
4/7/2016 13 Dr. Priya Mathew SJCE Mysore Mathematics Education
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.
4/7/2016 14 Dr. Priya Mathew SJCE Mysore Mathematics Education
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.
4/7/2016 15 Dr. Priya Mathew SJCE Mysore Mathematics Education
Theorem: If n2 is odd then n is odd
Proof : its contrapositive is
if n is not odd then n2 is not odd To prove contrapositive assume that n is not odd
Therefore n = m for some integer m therefore n2 = (2m)2 = 4 m2 = 2(2 m2 ) is even
Hence n2 is not odd
Therefore, If n2 is odd then n is odd
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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 4/7/2016 17 Dr. Priya Mathew SJCE Mysore Mathematics Education
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.
4/7/2016 18 Dr. Priya Mathew SJCE Mysore Mathematics Education
The sum of two odd numbers is odd
Counter Example , 7+5 =12
4/7/2016 19 Dr. Priya Mathew SJCE Mysore Mathematics Education
Mathematical induction is a mathematical
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
4/7/2016 20 Dr. Priya Mathew SJCE Mysore Mathematics Education
The basis (base case): prove that the statement holds for the first natural number n. Usually, n = 0 or n = 1, ra