2-2: Conditional Statements

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2-2: Conditional Statements Objectives Identify, write and analyze the truth value of conditional statements. Write the inverse, converse and contrapositive of a conditional statement.

Conditional Statement Vocabulary Sketch Things to remember 2-2 XXXX NOTATION: A statement that can be written in if p, then q form. Conditional Statement The “if” or p part of a conditional statement. Hypothesis The “then” or q part of the conditional statement. Conclusion NOTATION:  read “not” The negation of a true statement is false and a false statement is true. Negation

Converse Inverse Contrapositive Vocabulary Sketch Things to remember 2-2 XXXX NOTATION: Switch the “if” and “then” parts of conditional statement. Converse Negate the “if” and “then” in the conditional statement. Inverse Switch and negate the “if” and “then” in the conditional statement. Contrapositive

2-2: Conditional Statements If p, then q Hypothesis: If p Conclusion: then q p q

2-2: Conditional Statements If an animal is a Blue Jay, then it is a bird. Blue Jays Birds Hypothesis: Conclusion: If an animal is a Blue Jay Then it is a bird.

2-2: Conditional Statements If today is Thanksgiving, then today is Thursday. Thanksgiving Thursday Hypothesis: Conclusion: If today is Thanksgiving, Then today is Thursday.

2-2: Conditional Statements A number is a rational number if it is an integer. Rational number integer Hypothesis: Conclusion: If a number is an integer, Then it’s a rational number.

2-2: Conditional Statements Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Write a conditional statement from: Two angles that are complementary are acute. If two angles are complementary, then they are acute.

2-2: Conditional Statements A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. If the hypothesis is false, the conditional statement is true, regardless of the truth value of the conclusion. Remember!

2-2: Conditional Statements Determine if the conditional is true. If false, give a counterexample. If this month is August, the next month is September. True If two angles are acute, then they are congruent. False. Counterexample:________________ If an even number greater than 2 is prime, then 5 + 4 = 8. True. Hypothesis is false. If a number is odd, then it is divisible 3. False. Counterexample:_______________________________

2-2: Conditional Statements The negation of statement p is “not p,” written as ~p. The negation of a true statement is false, and the negation of a false statement is true.

2-2: Conditional Statements Determine which statement is the converse, inverse and contrapositive of the conditional statement. Use the science fact to find the truth value of each. Science Fact Adult insects have six legs. No other animals have six legs. If an animal is an adult insect, then it has six legs. If an animal is not an adult insect, then it doesn’t have six legs. p q True If an animal doesn’t have six legs, then it is not an adult insect. q p True If an animal has six legs, then it is an adult insect. q p True

2-2: Conditional Statements Determine which statement is the converse, inverse and contrapositive of the conditional statement. Find the truth value of each. If an animal is a cat, then it has four paws. If an animal doesn’t have four paws, then it isn’t a cat. True q p If an animal has four paws, then it is a cat. q p False If an animal is not a cat, then it doesn’t have four paws. p q False

2-2: Conditional Statements Logically Equivalent Statements: Related conditional statements that have the same truth value. Conditional Statement  Contrapositive Inverse  Converse The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive. Helpful Hint

2-2: Conditional Statements Assignment p. 84: 2-36 even