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2.1 Conditional Statements
Objectives
Recognize Conditional Statements and Their Parts.
Write Converses, Inverses, and Contrapositives of Conditional Statements.
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Definition
A conditional statement is a statement that is written, or that can be written, in “if-then” form. When a statement is in “if-then” form, the phrase that follows “if” is called the hypothesis, and the phrase that follows “then” is called the conclusion.
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Definition
If a figure is a pentagon, then it has five sides. If it is snowing, then it is cloudy.
Hypothesis
Conclusion
Hypothesis
Conclusion
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Symbols
When working with conditional statements, it is often handy to use shortcut notations. You may see any of the following: Hypothesis Conclusion if p then q p implies q p q
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5. Identifying the Hypothesis and the Conclusion
Identify the hypothesis (p) and the conclusion (q). a. If an animal is a turtle, then the animal is a reptile. Solution Hypothesis (p): an animal is a turtle Conclusion (q): the animal is a reptile
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6. Identifying the Hypothesis and the Conclusion
Identify the hypothesis (p) and the conclusion (q). b. If a number is even, then the number is not odd. Solution Hypothesis (p): a number is even Conclusion (q): the number is not odd
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7. Writing a Conditional Statement
A hypothesis (p) and a conclusion (q) are given. Use them to write a conditional statement, a. p: a figure is a square q: the figure is not a triangle Solution If a figure is a square, then the figure is not a triangle.
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8. Writing a Conditional Statement
A hypothesis (p) and a conclusion (q) are given. Use them to write a conditional statement, b. p: 9 is a perfect square q: 9 is not a prime number Solution If 9 is a perfect square, then 9 is not a prime number.
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9. Writing a Conditional Statement
Write the following statement in “if-then” form. Acute angles measure less than 90°. Solution If an angle is acute, then it measures less than 90°.
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10. Conditional Statements
A conditional statement may be either true or false. • A conditional statement is true if every time the hypothesis is true, then the conclusion is also true. • A conditional statement is false if there is a counterexample in which the hypothesis is true, but the conclusion is false.
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11. Is a Conditional Statement True or False?
Determine whether each conditional statement is true or false. a. If an angle measures 92°, then it is an obtuse angle. Solution This conditional statement is true. All angles that measure 92° are obtuse angles.
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12. Is a Conditional Statement True or False?
Determine whether each conditional statement is true or false. b. If a month begins with the letter J, then the month has 31 days. Solution This conditional statement is false. The month June starts with a J but has 30 days.
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Negation
The negation of a statement is formed by writing the negative of the statement. (Note: The notation for negation is =, so ~p is read “not p.”) Statement Negation The computer cover is red. The rarest blood group for humans is group AB.
The computer cover is not red.
The rarest blood group for humans is not group AB.
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15. Equivalent Statements
If two statements are both always true or both always false, we call them equivalent statements. • The conditional statement and the contrapositive statement in the table are both true and are examples of equivalent statements. • The converse statement and the inverse statement are both false and are examples of equivalent statements.
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16. Writing Related Conditional Statements
Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Converse: If it is cloudy, then it is raining. p q q p
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17. Writing Related Conditional Statements
Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Inverse: If it is not raining, then it is not cloudy. p q
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18. Writing Related Conditional Statements
Write the (a) converse, (b) inverse, and (c) contrapositive of the given conditional statement. If it is raining, then it is cloudy. Solution Contrapositive: If it is not cloudy, then it is not raining. p q
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