Conditional Statements Section 2–1 Geometry PreAP, Revised ©2013 viet.dang@humble.k12.tx.us 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Conditionals Chart – Page 2 Symbol Concept p Original Hypothesis q Original Conclusion → “Implies” ~ “Not” p → q “p implies q” “if p, then q” ~p “not p” Statement Symbols Conditional p → q Converse q → p Inverse ~p → ~q Contrapositive ~q → ~p 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Conditionals – Page 3 If p, then q Notation: p → q (if p then q) HYPOTHESIS – all words after “If” and before comma CONCLUSION – all words after “Then” 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Example 1 In this statement, “If a polygon is a hexagon, then it has exactly six sides” state the hypothesis and conclusion Subject Predicate A polygon is a hexagon is a polygon with six sides. If it is a hexagon, then it is a polygon with six sides. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 2 In this statement, “a number is a rational number if it is an integer.” State the hypothesis and conclusion. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 3 Rewrite this conditional statement in if-then form, “All 90° angles are right angles.” 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn Write an example of a conditional statement (if  then) about a topic in geometry. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Truth Values – Page 5 Determining whether a statement is TRUE or FALSE. True: To prove that the conclusion is true every time the hypothesis is satisfied. False: To prove if it is false, one has to find an example in which the conclusion is not true when the hypothesis is satisfied. A conditional is FALSE ONLY when hypothesis is true and conclusion is false. To show false you need a COUNTEREXAMPLE. Counterexample – example that proves a statement is false. Related conditional statements that have the same TRUTH VALUE are logically equivalent. Create a table (can all be T or all F) Conditional and Contrapositive always the same Converse and Inverse always the same 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 4 In this statement, “If this month is August, then next month is September” state whether the conditional statement is true. If it is false, provide a counterexample. When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 5 In this statement, “If n2 = 144, then n = 12” state whether the conditional statement is true. If it is false, provide a counterexample. When the hypothesis is true, the conclusion is also false because if n = –12, n2 also equals 144. So the conditional is false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “If it is February, then there are only 28 days in the month” state whether the conditional statement is true. If it is false, provide a counterexample. When the hypothesis is true, the conclusion is also false because if February can have 29 days in a leap year. So the conditional is false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Conditionals Chart – Page 2 Symbol Concept p Original Hypothesis q Original Conclusion → “Implies” ~ “Not” p → q “p implies q” “if p, then q” ~p “not p” Statement Symbols Conditional p → q Converse q → p Inverse ~p → ~q Contrapositive ~q → ~p 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Conditionals – Page 3 If p, then q Notation: p → q (if p then q) HYPOTHESIS – all words after “If” and before comma CONCLUSION – all words after “Then” 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Example 6 In this statement, “If a polygon is a hexagon, then it has exactly six sides” state the hypothesis and conclusion Subject Predicate A polygon is a hexagon is a polygon with six sides. If it is a hexagon, then it is a polygon with six sides. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 7 In this statement, “a number is a rational number if it is an integer.” State the hypothesis and conclusion. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 8 Rewrite this conditional statement in if-then form, “All 90° angles are right angles.” 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn Write an example of a conditional statement (if  then) about a topic in geometry. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Converse, Inverse, and Contrapositive – Page 4 A. Converse The CONVERSE is formed by REVERSING the hypotheses and conclusion. Notation: q → p (if q then p) B. Inverse The INVERSE is formed by NEGATING the hypotheses and conclusion. Notation: ~p → ~q (if not p then not q) C. Contrapositive The CONTRAPOSITIVE is formed by REVERSING and NEGATING the hypotheses and conclusion. Notation: ~q → ~p (if not q then not p) 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Example 9 In this statement, “If a polygon is a hexagon, then it has six sides” state the converse and determine whether it is true of false. If it is a hexagon, then it is a polygon with six sides. If a polygon has exactly six sides, then it is a hexagon. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 10 In this statement, “All squares are rectangles” state the converse and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “If an animal is an adult insect, then it has six legs” state the converse and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Example 11 In this statement, “If a polygon is a hexagon, then it has six sides” state the inverse and determine whether it is true of false. If it is a hexagon, then it is a polygon with six sides. If the polygon is NOT a hexagon, then it is NOT a polygon with six sides. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 12 In this statement, “If a number is even then it is divisible by two” state the converse and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “If an animal is an adult insect, then it has six legs” state the inverse and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

Example 13 In this statement, “If a polygon is a hexagon, then it has six sides” state the contrapositive and determine whether it is true of false. If it is a hexagon, then it is a polygon with six sides. If a polygon does NOT have exactly six sides, then it is NOT a hexagon. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 14 In this statement, “If you are a guitar player, then you are a musician,” state the contrapositive and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “If mA = 99°, then A is obtuse” state the contrapositive and determine whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 15 Fill out these statements below and identify whether it is true or false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “If I measure two angles which are acute, then they are congruent” state whether the conditional statement is true. If it is false, provide a counterexample. You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false. 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Assignment Page 77 8, 10-13 all, 16, 20-23, 25, 28, 31, 37, 38 2/23/2019 7:23 AM 2-1: Logic with Inductive Reasoning