Indirect Proof:

1
Section 2.2

• Indirect Proof
• Uses Laws of Logic to Prove Conditional
  Statements True or False

2
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.

3
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

4
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.

5
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.

6
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.

7
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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