5882270
Description
Mind Map by Danial Khir, updated more than 1 year ago
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Created by a deleted user
almost 8 years ago
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Copied by Danial Khir
almost 8 years ago
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PROOF
- Methods
- Indirect Method
- Contrapositive
- • Alternative to direct proof called contrapositive
proof.
- • Used to prove conditional statements of the form “
If P, then Q”
- • There are occasions where contrapositive
proof is much easier
- • There are occasions where contrapositive
proof is much easier
- • Used to prove conditional statements of the form “
If P, then Q”
- Steps of the
method :
- • P ->Q is logically equivalent to
~p->~q
- Steps 1: determine p and
q
- Steps 2:uses the proof by
contrapositive outline to
prove the statement.
- consequently 7 x + 9 = 2y + 1 is
odd
- Therefore 7 x + 9 is not
even
- the statement is then
proven
- the statement is then
proven
- Therefore 7 x + 9 is not
even
- consequently 7 x + 9 = 2y + 1 is
odd
- Steps 1: determine p and
q
- • P ->Q is logically equivalent to
~p->~q
- • Alternative to direct proof called contrapositive
proof.
- Contradiction
- • P ->Q is logically equivalent to ~p v q
- • ~(P ->Q) is logically equivalent
to p^~q
- Example 4
- STEP 1: write statement in symbol
- Step 2 : use proof by contradiction
outline to prove the statement
- X= 2k
- subtittute x= 2k into .
- contradiction is proven
- proven that
- statement is even
- the statement is then
proven
- the statement is then
proven
- statement is even
- proven that
- subtittute x= 2k into .
- Step 2 : use proof by contradiction
outline to prove the statement
- STEP 1: write statement in symbol
- • ~(P ->Q) is logically equivalent
to p^~q
- • P ->Q is logically equivalent to ~p v q
- Contrapositive
- Direct
Method
- Conditional Statement
- Outline
- Example
- If x is odd then x^2 is odd
- If x is odd then x^2 is odd
- Conditional Statement
- Mathematical
Induction
- Conditional statement
- Outline
- Inductive step
- Inductive step
- Basic step
- Example
- Conditional statement
- Indirect Method
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