Section 2.21 Indirect Proof: Uses Laws of Logic to Prove Conditional Statements True or False

Section 2.22 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.”

Section 2.23 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, the Tom lives in San Diego.

Section 2.24 Conditional and its Contrapositive The Law of Negative Inference 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

Section 2.25 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. 82. P → Q If Erin gets paid, she will go to the concert ~Q Erin didn’t go to the concert ∴ ~P Erin didn’t get paid. Strategy: 1.Suppose that ~Q is true. 2.Reason from the supposition until you reach a contradiction. 3.Note that the supposition claiming that ~Q is true must be false and that Q therefore must be true.

Section 2.26 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. Ex. 5 p. 84 / / / /