Section 2-2: Biconditional and Definitions TPI 32C: Use inductive and deductive reasoning

1. Section 2-2: Biconditional and Definitions TPI 32C: Use inductive and deductive reasoning to make conjectures • Objectives: • Write the inverse and contrapositive of conditional statements • Write Biconditionals and recognize good definitions Recall

2. Forms of a Conditional Statement Converse Inverse Contrapositive Biconditional

3. Symbolic Negation (~p OR ~q) • Negation of a statement has the opposite truth value. Statement: ABC is an obtuse angle. Negation: ABC is not an obtuse angle. Statement: Lines m and n are not perpendicular Negation: Lines m and n are perpendicular. Use negation for INVERSE! ~

4. Form of a Conditional Statement Symbol ~ is used to indicate the word “NOT” Inverse States the opposite of both the hypothesis and conclusion. (~p~q) If not p, then not q. Conditional: pq :Iftwo angles are vertical, thenthey are congruent. Inverse: ~p~q:Iftwo angles are not vertical, thenthey are not congruent.

5. Inverse • Inverse of a conditional negates BOTH the hypothesis and conclusion. Conditional If a figure is a square, then it is a rectangle. NEGATE BOTH Inverse If a figure is NOT a square, then it is NOT a rectangle.

6. Form of a Conditional Statement Contrapositive (~q~p) If not q, then not p. Conditional: pq : Iftwo angles are vertical, thenthey are congruent. Contrapositive: ~q~p:Iftwo angles are not congruent, thenthey are not vertical. Switch the hypothesis and conclusion & state their opposites. (~q~p) (Do Converse and Inverse)

7. Contrapositive • Contrapositive switches hypothesis and conclusion AND negates both. • A conditional and its contrapositive are equivalent. They have the same truth value). Conditional If a figure is a square, then it is a rectangle. SWITCH AND NEGATE BOTH Contrapositive If a figure is NOT a rectangle, then it is NOT a square.

8. Lewis Carroll, the author of Alice's Adventures in Wonderland and Through the Looking Glass, was actually a mathematics teacher.   As a hobby, Carroll wrote stories that contain amusing examples of logic.  His works reflect his passion for mathematics Lewis Carroll’s “Alice in Wonderland” quote: "You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!" Translate into a conditional: If I am sleeping, then I am breathing. If I am not sleeping, then I am not breathing. Inverse of a conditional: If I am not breathing, then I am not sleeping. Contrapositive of a conditional:

9. Form of a Conditional Statement Biconditional p  q • Write a bi-conditional only ifBOTH the conditional and the converse are TRUE. • Connect the conditional & its converse with the word “and” • Write by joining the two parts of each conditional with the phrase “if and only if” of “iff” for shorthand. • Symbolically: p  q

10. Bi-conditional Statements Conditional Statement: If two angles have the same measure, then the angles are congruent. Converse: If two angles are congruent, then they have the same measure. Both statements are true, so…. …you can write a Biconditional statement: Two angles have the same measure if and only if the angles are congruent.

11. Try It! Write a Bi-conditional Statement Consider the following true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If x = 5, then x + 15 = 20. Converse: If x + 15 = 20, then x = 5. Since both the conditional and its converse are true, you can combine them in a true biconditional using the phrase if and only if. Biconditional: x = 5 if and only if x + 15 = 20.

12. Separate a Biconditional • Write a biconditional as two conditionals that are converses of each other. Consider the biconditional statement: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Statement 1: If a number is divisible by 3, then the sum of its digits is divisible by 3. Statement 2: If the sum of a numbers digits is divisible by 3, then the number is divisible by 3.

13. Try It! Separate a Biconditional Write the two statements that form this biconditional. Biconditional: Lines are skew if and only if they are noncoplanar. Conditional: If lines are skew, then they are noncoplanar. Converse: If lines are noncoplanar, then they are skew.

14. Writing Definitions as Biconditionals • Good Definitions: • Help identify or classify an object • Uses clearly understood terms • Is precise avoiding words such as sort of and some • Is reversible, meaning you can write a good definition as a biconditional (both conditional and converse are true) Show definition of perpendicular lines is reversible Definition: Perpendicular lines are two lines that intersect to form right angles Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse If two lines intersect to form right angles, then they are perpendicular. Since both are true converses of each other, the definition can be written as a true biconditional: “Two lines are perpendicular iff they intersect to form right angles.”

15. Writing Definitions as Biconditionals Show that the definition of triangle is reversible. Then write it as a true biconditional. Steps 1. Write the conditional 2. Write the converse 3. Determine if both statements are true 4. If true, combine to form a biconditional. Definition: A triangle is a polygon with exactly three sides. True Conditional: If a polygon is a triangle, then it has exactly three sides. True Converse: If a polygon has exactly three sides, then it is a triangle. Biconditional: A polygon is a triangle if and only if it has exactly three sides.

16. Writing Definitions as Biconditionals Is the following statement a good definition? Explain. An apple is a fruit that contains seeds. Conditional: If a fruit is an apple then it contains seeds. Converse: If a fruit contains seed then it is an apple. There are many other fruits containing seeds that are not apples, such as lemons and peaches. These are counterexamples, so the reverse of the statement is false. The original statement is not a good definition because the statement is not reversible.

17. Summary