1 If-then Statements “If-then statements” are also called “Conditional Statements” or just “Conditionals” “If-then statements” are also called “Conditional Statements” or just “Conditionals”

2 Examples of Conditionals “If it rains after school, then I will give you a ride home.” “If B is between A and C, then AB + BC = AC.” Basic Form: If p, then q. p represents the hypothesis q represents the conclusion “If it rains after school, then I will give you a ride home.” “If B is between A and C, then AB + BC = AC.” Basic Form: If p, then q. p represents the hypothesis q represents the conclusion

3 Hypothesis & Conclusion “If it rains after school, then I will give you a ride home.” The hypothesis is “it rains after school” The conclusion is “I will give you a ride home” “If it rains after school, then I will give you a ride home.” The hypothesis is “it rains after school” The conclusion is “I will give you a ride home”

4 Hypothesis & Conclusion “If B is between A and C, then AB + BC = AC.” The hypothesis is “B is between A and C” The conclusion is “AB + BC = AC” “If B is between A and C, then AB + BC = AC.” The hypothesis is “B is between A and C” The conclusion is “AB + BC = AC”

5 Other forms of Conditionals General Form Examples: If p, then q. If x = 5, then x 2 = 25. p implies q. x = 5 implies x 2 = 25. p only if q. x = 5 only if x 2 = 25. q if p. x 2 = 25 if x = 5. General Form Examples: If p, then q. If x = 5, then x 2 = 25. p implies q. x = 5 implies x 2 = 25. p only if q. x = 5 only if x 2 = 25. q if p. x 2 = 25 if x = 5.

6 Find the hypothesis & the conclusion Example 1: 3x = 12 implies x = 4. hypothesis: 3x = 12. conclusion: x = 4. Example 2: You can go out if you do all your homework. hypothesis: you do all your homework conclusion: you can go out Example 1: 3x = 12 implies x = 4. hypothesis: 3x = 12. conclusion: x = 4. Example 2: You can go out if you do all your homework. hypothesis: you do all your homework conclusion: you can go out

7 Converses The converse of a conditional is formed by interchanging the hypothesis and the conclusion. Statement: If p, then q. Converse: If q, then p. The converse of a conditional is formed by interchanging the hypothesis and the conclusion. Statement: If p, then q. Converse: If q, then p.

8 Example of writing a converse Sometimes a statement and its converse are both true. Statement: If 4x = 20, then x = 5. Converse: If x = 5, then 4x = 20. Sometimes a statement and its converse are both true. Statement: If 4x = 20, then x = 5. Converse: If x = 5, then 4x = 20.

9 Another example of a converse Sometimes a statement is true and its converse is false. Statement: If Ed lives in Texas, then he lives south of Canada. Converse: If Ed lives south of Canada, then he lives in Texas. Sometimes a statement is true and its converse is false. Statement: If Ed lives in Texas, then he lives south of Canada. Converse: If Ed lives south of Canada, then he lives in Texas.

10 Counterexamples A conditional is false if an example can be found for which the hypothesis is true and the conclusion is false. Such an example is called a counterexample. It takes only ONE counterexample to disprove a statement. A conditional is false if an example can be found for which the hypothesis is true and the conclusion is false. Such an example is called a counterexample. It takes only ONE counterexample to disprove a statement.

11 Finding a counterexample Statement: If Ed lives in Texas, then he lives south of Canada. Converse: If Ed lives south of Canada, then he lives in Texas. The converse is false. Find one counterexample that proves the converse is false. Statement: If Ed lives in Texas, then he lives south of Canada. Converse: If Ed lives south of Canada, then he lives in Texas. The converse is false. Find one counterexample that proves the converse is false.

12 Finding a counterexample True Statement: If x = 7, then x < 10. Converse: If x < 10, then x = 7. (This converse is false.) Find one counterexample that proves the converse is false. True Statement: If x = 7, then x < 10. Converse: If x < 10, then x = 7. (This converse is false.) Find one counterexample that proves the converse is false.

13 Biconditionals If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. A statement that contains the words “if and only if” is called a biconditional. General form: p if and only if q. If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. A statement that contains the words “if and only if” is called a biconditional. General form: p if and only if q.

14 Writing a Biconditional Statement: If 4x = 20, then x = 5. Converse: If x = 5, then 4x = 20. Since both of these are true, they can be written into one biconditional: 4x = 20 if and only if x = 5. Statement: If 4x = 20, then x = 5. Converse: If x = 5, then 4x = 20. Since both of these are true, they can be written into one biconditional: 4x = 20 if and only if x = 5.

15 Writing another Biconditional Statement: If today is Monday, then tomorrow is Tuesday. Converse: If tomorrow is Tuesday, then today is Monday. Since both of these are true, they can be written into one biconditional: Today is Monday if and only if tomorrow is Tuesday. Statement: If today is Monday, then tomorrow is Tuesday. Converse: If tomorrow is Tuesday, then today is Monday. Since both of these are true, they can be written into one biconditional: Today is Monday if and only if tomorrow is Tuesday.

16 Write TWO Conditionals from ONE Biconditional Biconditional: x + 5 = 9 if and only if x = 4. If x + 5 = 9, then x = 4. If x = 4, then x + 5 = 9. Biconditional: x + 5 = 9 if and only if x = 4. If x + 5 = 9, then x = 4. If x = 4, then x + 5 = 9.

17 Write TWO Conditionals from ONE Biconditional Biconditional: Segments are congruent if and only if they have the same length. If segments are congruent, then they have the same length. If segments have the same length, then they are congruent. Biconditional: Segments are congruent if and only if they have the same length. If segments are congruent, then they have the same length. If segments have the same length, then they are congruent.

18 Homework: Prepare for a quiz on 1-5 & 2-1 and Honors: Page 35 #1-6 all, 8-30 even CP: Page 35 #1-6 all, 8-12 even, all, 25, 28 Prepare for a quiz on 1-5 & 2-1 and Honors: Page 35 #1-6 all, 8-30 even CP: Page 35 #1-6 all, 8-12 even, all, 25, 28