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Conditional Statements. BIG IDEA: REASONING AND PROOF ESSENTIAL UNDERSTANDINGS: Some mathematical relationships can be described using a variety of if-then statements. Each conditional statement has a converse, an inverse, and a contrapositive. MATHEMATICAL PRACTICE: Attend to precision.
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Conditional Statements BIG IDEA: REASONING AND PROOF ESSENTIAL UNDERSTANDINGS: Some mathematical relationships can be described using a variety of if-then statements. Each conditional statement has a converse, an inverse, and a contrapositive. MATHEMATICAL PRACTICE: Attend to precision
Getting Ready • The company that prints the bumper sticker at the left below accidentally reworded the original statement and printed the sticker three different ways. Suppose the original bumper sticker is true. Are the other bumper stickers true or false? Explain. • If you can read this, a) If you are too close, then you are too close then you can read this Original b) If you cannot read this, then you are not too close c) If you are not too close, then you cannot read this
Background • Postulates and theorems in Geometry are written as conditional statements, and for that reason, it is important for students to understand these types of statements. You will encounter many geometric definitions as you progress through the textbook, and each of these definitions is a true conditional statement whose converse is also true.
Conditional Statements • A Conditional is an _______-_____________ statement • The Hypothesis is the part ______ following ______ • The Conclusion is the part ______ following ______ • The Truth value of a statement is “__________” or “__________” according to whether the statement is true or false, respectively. • 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, if you find one counterexample for which the hypothesis is true and the conclusion is false, then the truth value of the conditional is false.
Examples • 1. What are the hypothesis and the conclusion of the conditional? • If a number is even, then it is divisible by 2. • 2 How can you write the following statement as a conditional? • An even number greater than two is not prime.
Examples • 3. Is the conditional true or false? If it is false, find a counterexample. • a) If you live in Miami, then you live in Florida. • b) If a number is divisible by 5, then it is odd. • c) If a month has 28 days, then it is February.
Related Conditional Statements • Converse: __________________ the hypothesis and the conclusion of the conditional • The Negation of a statement has the _______________ meaning of the original statement. • Inverse: ____________ both the hypothesis and the conclusion of the conditional • Contrapositive: ______________ and _______________ both the hypothesis and the conclusion of the conditional • Equivalent statements: have the __________ truth value
Example • 4. What are the converse, inverse, and contrapositive of the conditional statement below? What are the truth values of each? If a statement is false, give a counterexample. • If a figure is a rectangle, then it is a parallelogram.
Example • 5. What are the converse, inverse, and contrapositive of the conditional statement below? What are the truth values of each? If a statement is false, give a counterexample. • If the temperature outside is below freezing, then ice can form on the sidewalks.
