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 FalseFor all integers n, is positive.

2 complementary angles cannot be .

3n

LogicInductive 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 Conjecture1. 1, 2, 4, 7, 11, _____

2. Jan, March, May, ______

• 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→qIf I work hard, then I will succeed.

Converse If q, then p. Flip q→pIf I will succeed, then I worked hard.

Conditional If p, then q. p→qIf I work hard, then I will succeed.

Inverse If not p, then not q negate ~p→~qIf I don’t work hard, then I will not succeed.

Contrapositive If not q, then not p flip & negate ~q→~pIf I don’t succeed, then I didn’t work hard.

True or FalseThat 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 pattern2. Make a conjecture3. Prove or find a counterexample

Prove or find a counterexample

For all integers n, is positive.

2 complementary angles cannot be .

3n

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 → qT 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. TNote --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 → ~qIf m<A ≠30, then <A is not acute.

Contrapositive ~q → ~pIf <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 → ~qIf 2 angles are not vertical, then they are not

Contrapositive ~q → ~pIf 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.