Direct Proof by Contradiction in Mathematics
Section 2.6
Direct Proof by Contradiction
Review: Proof by contraposition
•
Express the statement in the form:
–
x , if P(x) then Q(x)
•
Rewrite this statement in the contrapositive
form:
–
x , if not Q(x) then not P(x)
•
Prove the contrapositive in a direct proof
You try
•
Prove: For all integers if x – y is odd, the x
is odd or y is odd.
More practice
Prove:
If x and y are two integers for which
x+y is even, then x and y have the same
parity
.
•
DEFINITION - Two integers are said to
have the same parity if they are both odd or
both even.
Prove: If the sum of two real numbers is less
than 50, then at least one of the numbers is
less than 25.
Proof by Contradiction
•
Another tool for “direct proof” is Proof by
Contradiction.
•
Introduction:
If A is true, what do you know about
¬ A ?
If A is false, what do you know about ¬ A ?
If ¬ A is false, what do you know about A?
Proof by Contradiction
1.
Suppose that the statement is false. That
is, suppose the negation of the statement is
true
2.
Show that this new supposition leads to a
contradiction.
3.
Conclude that the original statement must
be true.
Examples of this technique
•
There is no greatest integer
Examples of this technique
•
There is no integer that is both even and odd
Examples of this technique
•
If 3n+2 is odd, then n is odd
•
[Hint, be careful writing the negation of this
statement].
Learn about direct proof by contradiction in mathematics, including the process, examples, and applications. Understand how to prove statements by assuming their negations and deriving a contradiction to establish the original statement's truth.
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Presentation Transcript
Section 2.6 Direct Proof by Contradiction
Review: Proof by contraposition Express the statement in the form: x , if P(x) then Q(x) Rewrite this statement in the contrapositive form: x , if not Q(x) then not P(x) Prove the contrapositive in a direct proof
You try Prove: For all integers if x y is odd, the x is odd or y is odd.
More practice Prove: If x and y are two integers for which x+y is even, then x and y have the same parity. DEFINITION - Two integers are said to have the same parity if they are both odd or both even. Prove: If the sum of two real numbers is less than 50, then at least one of the numbers is less than 25.
Proof by Contradiction Another tool for direct proof is Proof by Contradiction. Introduction: If A is true, what do you know about A ? If A is false, what do you know about A ? If A is false, what do you know about A?
Proof by Contradiction 1. Suppose that the statement is false. That is, suppose the negation of the statement is true 2. Show that this new supposition leads to a contradiction. 3. Conclude that the original statement must be true.
Examples of this technique There is no greatest integer
Examples of this technique There is no integer that is both even and odd
Examples of this technique If 3n+2 is odd, then n is odd [Hint, be careful writing the negation of this statement].
