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Section 2.2

- Indirect Proof
- Uses Laws of Logic to Prove Conditional

Statements True or False

Forms of Indirect Proof

- Conditional (or Implication)
- P ? Q
- If it is a wheel, then it is round.
- Converse of Conditional
- Q ?P
- if it is round, then it is a wheel.
- Inverse of Conditional
- P ? Q
- If it is not a wheel, then it is not round.
- Contrapositive of Conditional
- Q ? P
- If it is not round, then it is not a wheel.

Conditional and its Contrapositive

- The contrapositive of a given conditional is

always true when the conditional is true. - A conditional statement can always be replaced

with its contrapositive. - Conditional
- If two angles are supplementary, then the sum of

the angles is 180?. - Contrapositive
- If the sum of two angles is not 180?, then the

two angles are not supplementary

Conditional and its Inverse and Converse

- In general, the inverse and converse of a given

conditional need not be true when the conditional

is true. - Conditional
- If Tom lives in San Diego, then Tom lives in

California. - Inverse
- If Tom does not live in San Diego, then Tom does

not live in California. - Converse
- If Tom lives in California, then Tom lives in San

Diego.

Indirect Proof Law of Negative Inference

(Contraposition)

- Although direct proofs (2-column) are the most

common type of proofs, some theorems are more

easily proved using the format of an indirect

proof. - p. 81.
- P Q If Erin gets paid, she

will go to the concert - Q Erin didnt go to the concert
- P Erin didnt get paid.

Method of Indirect Proof

- To prove the statement P?Q or to complete the

proof problem of the form - Given P
- Prove Q
- where Q may be a negation, use the following

steps - Suppose that Q is true.
- Reason from the supposition until you reach a

contradiction. - Note that the supposition claiming that Q is

true must be false and that Q must therefore be

true. - Step 3 completes the proof.

Example of Indirect Proof

- Prove If two lines are cut by a transversal so

that corresponding angles are not congruent, then

the two lines are not parallel. - Given
- r and s are cut by transversal t.
- ?1 ? ?5
- Prove r s
- Assume that r s When they are cut
- by the transversal, corresponding
- angles are congruent. But ?1 ? ?5
- by hypothesis. Thus the assumed
- statement that r s is false. It follows
- that r s .

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