Introduction to Deductive Proofs 1.4 Introduction to Deductive Proofs Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definition A conditional statement is a statement that is written, or that can be written, in “if-then” form. When a statement is in “if-then” form, the phrase that follows “if” is called the hypothesis, and the phrase that follows “then” is called the conclusion. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definition If a figure is a pentagon, then it has five sides. If it is snowing, then it is cloudy. Hypothesis Conclusion Hypothesis Conclusion Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Symbols When working with conditional statements, it is often handy to use shortcut notations. You may see any of the following: Hypothesis Conclusion if p then q p implies q p q Copyright © 2014 Pearson Education, Inc.

Identifying the Hypothesis and the Conclusion Identify the hypothesis (p) and the conclusion (q). a. If an animal is a turtle, then the animal is a reptile. Solution Hypothesis (p): an animal is a turtle Conclusion (q): the animal is a reptile Copyright © 2014 Pearson Education, Inc.

Identifying the Hypothesis and the Conclusion Identify the hypothesis (p) and the conclusion (q). b. If a number is even, then the number is not odd. Solution Hypothesis (p): a number is even Conclusion (q): the number is not odd Copyright © 2014 Pearson Education, Inc.

Writing a Conditional Statement A hypothesis (p) and a conclusion (q) are given. Use them to write a conditional statement, a. p: a figure is a square q: the figure is not a triangle Solution If a figure is a square, then the figure is not a triangle. Copyright © 2014 Pearson Education, Inc.

Writing a Conditional Statement A hypothesis (p) and a conclusion (q) are given. Use them to write a conditional statement, b. p: 9 is a perfect square q: 9 is not a prime number Solution If 9 is a perfect square, then 9 is not a prime number. Copyright © 2014 Pearson Education, Inc.

Writing a Conditional Statement Write the following statement in “if-then” form. Acute angles measure less than 90°. Solution If an angle is acute, then it measures less than 90°. Copyright © 2014 Pearson Education, Inc.

Conditional Statements A conditional statement may be either true or false. • A conditional statement is true if every time the hypothesis is true, then the conclusion is also true. • A conditional statement is false if there is a counterexample in which the hypothesis is true, but the conclusion is false. Copyright © 2014 Pearson Education, Inc.

Is a Conditional Statement True or False? Determine whether each conditional statement is true or false. a. If an angle measures 92°, then it is an obtuse angle. Solution This conditional statement is true. All angles that measure 92° are obtuse angles. Copyright © 2014 Pearson Education, Inc.

Is a Conditional Statement True or False? Determine whether each conditional statement is true or false. b. If a month begins with the letter J, then the month has 31 days. Solution This conditional statement is false. The month June starts with a J but has 30 days. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Negation The negation of a statement is formed by writing the negative of the statement. (Note: The notation for negation is =, so ~p is read “not p.”) Statement Negation The computer cover is red. The rarest blood group for humans is group AB. The computer cover is not red. The rarest blood group for humans is not group AB. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc.

Equivalent Statements If two statements are both always true or both always false, we call them equivalent statements. • The conditional statement and the contrapositive statement in the table are both true and are examples of equivalent statements. • The converse statement and the inverse statement are both false and are examples of equivalent statements. Copyright © 2014 Pearson Education, Inc.

Writing Related Conditional Statements Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Converse: If it is cloudy, then it is raining. p q q p Copyright © 2014 Pearson Education, Inc.

Writing Related Conditional Statements Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Inverse: If it is not raining, then it is not cloudy. p q Copyright © 2014 Pearson Education, Inc.

Writing Related Conditional Statements Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Contrapositive: If it is not cloudy, then it is not raining. p q Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Deductive Reasoning Deductive reasoning is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved true by deductive reasoning is called a theorem. Copyright © 2014 Pearson Education, Inc.

Giving Reasons for Statements Solve 5x + 12 = 47. Give a reason to justify each statement. Solution Statements Reasons Given Subtraction Property of Equality Division Property of Equality Copyright © 2014 Pearson Education, Inc.

Giving Reasons for Statements Solve 7x – 10(5 + 3x) = 2x. Give a reason to justify each statement. Solution Statements Reasons 7x – 10(5 + 3x) = 2x Given Multiply or Distributive Property Simplify or Combine like terms Addition Property of Equality Division Property of Equality Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Proof Basics A proof is an argument that uses logic to establish the truth of a statement. There are many formats for proofs, but for now, we will use a two-column proof. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Writing a Direct Proof Write a direct proof. Given: Given Prove: Conclusion Proof: Statements Reasons Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Writing a Direct Proof Page 32 Exercise 2 Copyright © 2014 Pearson Education, Inc.