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2.2 Conditional Statements Goal: Students will be able: To recognize conditional statements and their parts. To write converses, inverses, and contrapositives of conditional statements. Conditional Statement A statement that can be written in “if – then” form. Symbol: p → q, read if p then q, or p implies q. Example: If it rains on Thursday, then the baseball game will be canceled. q p p Hypothesis The phrase immediately following the word if in a conditional statement The p part following if. Conclusion The phrase immediately following the word then in a conditional statement. If p, then q. Hypothesis Conclusion Examples 1: Determine Hypothesis and conclusion If trout are fish, then trout live in a pond. Hypothesis: trout are fish Conclusion: trout live in a pond If you buy a car, then you get $1500 cash back. Hypothesis: you buy a car Conclusion: you get $1500 cash back Got it 1? Identify the hypothesis and conclusion of each statement. If an animal is a robin, then the animal is a bird. Hypothesis: an animal is a robin Conclusion: the animal is a bird. If an angle measures 180°, then the angle is obtuse. Hypothesis: an angle measures 180° Conclusion: the angle is obtuse If a polygon has 6 sides, then it is a hexagon. Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon Example 2: Writing a Conditional How can you write the following statement as a conditional? Vertical angles share a vertex. Step 1: Identify the hypothesis and conclusion. In order for two angles to be vertical, they must share a vertex. So the set of vertical angles is inside the set of angles that share a vertex. Hypothesis: Vertical angles Conclusion: share a vertex. If two angles are vertical angles, then they share a vertex. Got it 2? How can you write “Dolphins are mammals” as a conditional? If an animal is a dolphin, then it is a mammal. Mammals Dolphins Truth Value Is either true or false To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. To show that a conditional is false, find only one counterexample, where the hypothesis is true, and the conclusion is false. Example 3: Finding the truth value of a conditional. Is the conditional true or false? If it is false, find a counterexample. If a number is divisible by 3, then it is odd. The conclusion is false. The number 12 is divisible by 3, and 12 is even. Got it? 3: Finding the truth value of a conditional. Is the conditional true or false? If it is false, find a counterexample. If a month has 28 days, then it is February. False, January has 28 days plus 3 more. If two angles form a linear pair, then they are supplementary. True Negation Is the opposite of the original statement ~p, read not p Symbol: ~ Examples: p: The sky is blue. ~p: The sky is not blue. q: A triangle has 4 sides. ~q: A triangle does not have 4 sides. Converse, Inverse, and Contrapositive Statement Formed by Conditional Given hypothesis and conclusion Converse Exchange the hypothesis and conclusion of the conditional Inverse Contrapositive Negate both the hypothesis and conclusion of the conditional Exchange and Negating both the hypothesis and conclusion of the conditional Symbols Examples p→q If two angles have the same measure, then they are congruent. q→p If two angles are congruent, then they have the same Measure. ~p → ~q If two angles do not have the same measure, then they are not congruent. ~q → ~p If two angles are not congruent, then they do not have the same measure. Equivalent Statements Have the same truth value The conditional and the contrapositive are equivalent statements. The converse and the inverse are equivalent statements. Biconditional Statements: When a conditional statement and its converse are both true, you can write them as a single biconditional statement. A biconditional statement is a statement that contains the phrase “if and only if.” Any valid definition can be written as a biconditional statement. Example 1: Rewrite the conditional statement in if-then form. All birds have feathers. Conditional: If it is a bird, then it has feathers. Two angles are supplementary if they are a linear pair. Conditional: If two angles are a linear pair, then they are supplementary. Example 2: Write the converse, inverse, and contrapositive of the following conditional? If a dog is a Great Dane, then it is large. Converse: If the dog is large, then it is a Great Dane. Inverse: If the dog is not a Great Dane, then it is not large. Contrapositive: If the dog is not large, then it is not a Great Dane. Example 4. Write the definition of perpendicular lines as a biconditional. Definition: If two lines intersect to form a right angle, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Biconditional: Lines intersect to form right angles iff they are perpendicular lines.