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 Conditional Statements and Converses StatementExampleSymboli c You read as ConditionalIf an angle is a straight angle, then its measure is 180º. p  qIf p, then q. ConverseIf the measure of an angle is 180º, then it is a straight angle. q  pIf q then p.

Forms of a Conditional Statement Converse Inverse Contrapositive Biconditional

Symbolic Negation (~p  ~q) Statement:  ABC is an obtuse angle. Negation of a statement has the opposite truth value. Negation:  ABC is not an obtuse angle. Statement: Lines m and n are not perpendicular Negation: Lines m and n are perpendicular.

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

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

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

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

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. Inverse of a conditional: If I am not sleeping, then I am not breathing. Contrapositive of a conditional: If I am not breathing, then I am not sleeping. 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

Form of a Conditional Statement Write a bi-conditional only if BOTH 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 p  q

Bi-conditional Statements Conditional Statement: If two angles 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.

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. Write a Bi-conditional Statement

Separate a Biconditional Consider the biconditional statement: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Write a biconditional as two conditionals that are converses of each other. 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.

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

Writing Definitions as Biconditionals Show definition of perpendicular lines is reversible 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) Definition: Perpendicular lines are two lines that intersect to form right angles 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.” 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.

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

Is the following statement a good definition? Explain. Conditional: If a fruit is an apple then if 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. An apple is a fruit that contains seeds. Writing Definitions as Biconditionals

StatementExampleSymbolicYou read as ConditionalIf an angle is a straight angle, then its measure is 180º. p  qIf p, then q. ConverseIf the measure of an angle is 180º, then it is a straight angle q  pIf q then p. InverseIf an angle is not a straight angle, then its measure is not 180. ~p  ~qIf not p, then not q ContrapositiveIf an angle does not measure 180, then the angle is not a straight angle. ~q  ~pIf not q, then not p. BiconditionalAn angle is a straight angle if and only if its measure is 180º. p  qp if and only if q. P iff q