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Get Ready To Be Logical! • 1. Books and notebooks out. • 2. Supplies ready to go. • 3. Answer the following: The sum of 2 positive integers is ___________ True or False 3 n For all integers n, is positive. 2 complementary angles cannot be . Logic Inductive Reasoning is a process of reasoning that a rule or statement is true because of a specific case that is usually drawn from a pattern or observation. Examples 1. 1, 2, 4, 7, 11, _____ 2. Jan, March, May, ______ Conjecture • Deductive Reasoning is a process of using logic to draw conclusions using definitions, facts, or properties. This may include postulates and theorems. Conditional If p, then q. p is hypothesis q is conclusion p→q If I work hard, then I will succeed. Converse If q, then p. Flip q→p If I will succeed, then I worked hard. Conditional If p, then q. If I work hard, then I will succeed. p→q Inverse If not p, then not q negate ~p→~q If I don’t work hard, then I will not succeed. Contrapositive If not q, then not p flip & negate ~q→~p If I don’t succeed, then I didn’t work hard. True or False That is the Question??? To prove a statement false, one must provide a counterexample. A counterexample is a drawing, statement or number. The counterexample must satisfy the hypothesis but fail to satisfy the conclusion. 1. Look for a pattern 2. Make a conjecture 3. Prove or find a counterexample Prove or find a counterexample 3 For all integers n, n is positive. 2 complementary angles cannot be . Truth value-- is true in all situations except when hypothesis is true and the conclusion is false. p = you make an A q = I will buy you a car p q p→q T T You made an A, then I bought the car. T T F You made an A, but I did not buy the car. F F T You did not make an A, but I bought the car anyway. T F F You did not make an A, then I did not buy the car. T Note --satisfying the hypothesis and failing to satisfy the conclusion is what a counterexample does. Note—counterexamples are used to prove statements false! Write the converse, inverse, and contrapositive of the following. State if true or false. If false give counterexample. If m<A = 30, then <A is acute. If m<A = 30, then <A is acute. p → q Converse q → p If <A is acute, then m<A = 30. Inverse ~p → ~q If m<A ≠30, then <A is not acute. Contrapositive ~q → ~p If <A is not acute, then m<A ≠ 30. Write the converse, inverse, and contrapositive of the following. State if true or false. If false give counterexample. If 2 angles are vertical, then they are If 2 angles are vertical, then they are p → q Converse q → p If 2 angles are , then they are vertical. Inverse ~p → ~q If 2 angles are not vertical, then they are not Contrapositive ~q → ~p If 2 angles are not , then they are not vertical. Biconditionals • A biconditional is a blending of the conditional statement and the converse. • A biconditional may only be written if BOTH the conditional AND the converse are true. • A biconditional is in the form of p if and only if q. This may be abbreviated p iff q or p q. • The statement --If an angle is a right angle, then it measures 90 ° This statement is true. • The converse—If an angle measures 90°, then it is a right angle. This converse is true. • Therefore, we can write a biconditional. An angle measures 90° if and only if it is a right angle.