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Section 2- 2, Part 2: Inverse & Contrapositive. Objectives: Write the inverse and contrapositive of conditional statements. Recall. Forms of a Conditional Statement. Converse. Inverse. Contrapositive. Biconditional. Symbolic Negation (~p OR ~q).
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Section 2-2, Part 2: Inverse & Contrapositive • Objectives: • Write the inverse and contrapositive of conditional statements Recall
Forms of a Conditional Statement Converse Inverse Contrapositive Biconditional
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! ~
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: pq :Iftwo angles are vertical, thenthey are congruent. Inverse: ~p~q:Iftwo angles are not vertical, thenthey are not congruent.
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.
Form of a Conditional Statement Contrapositive (~q~p) If not q, then not p. Conditional: pq : 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)
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.
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:
