LG.1.G.1 Define, compare and contrast inductive reasoning and deductive reasoning for making predictions based on real world situations, Venn

diagrams, matrix logic, conditional statements (statement, inverse, converse, and contrapositive), figural patterns

LG.1.G.6 Give justification for conclusions reached by deductive reasoning

3.1 Conditional Statements, Converses, Inverses,

Contrapositives

Conditional StatementsIf . . . Then

Conditional Statement – A compound statement formed by combining two sentences (or facts) using the words “if…then.”

Example: If you pass all of your tests, then you will pass the class.

Definitions

Hypothesis: The part of the sentence that follows the word “IF”

Conclusion: The part of the sentence that follows the word “THEN”

EXAMPLE: If you do your homework on time, then you will have a better grade.

What is the hypothesis?you do your homework on time

What is the conclusion?you will have a better grade

It may be necessary to rewrite a sentence so that it is in conditional form (“if” first, “then” second).

Example:All surfers like big waves.Rewrite as a conditional statement.

If you are a surfer, then you like big waves.

Truth Value

When you determine whether a conditional statement is true or false, you determine its truth value.

Counterexamples

Counterexample: An example that proves a statement false

Write a counterexample for the following conditional statement:

If a student likes math, then he likes chemistry.

Converse

Converse of a Conditional Statement: Formed by interchanging the hypothesis and conclusion of the original statement.

Example:Conditional: If the space shuttle was launched,

then a cloud of smoke was seen.

Converse:If a cloud of smoke was seen, then the space shuttle was launched.

HINT

Try to associate the logical CONVERSE with Converse sneakers – think of the two parts of the sentence “putting on their sneakers and running to their new positions.”

Inverse

Inverse of a Conditional Statement: formed by negating the hypothesis and negating the conclusion of the original statement.

Put “not” into the hypothesis and the conclusion

Example

Conditional: If you grew up in Alaska, then you have seen snow.

Inverse:If you did not grow up in Alaska, then you have not seen snow.

HINT: Hint: To create an INverse, you need to INsert the word not into both the hypothesis and the conclusion.

Does the truth value of an inverse have to be the same as the truth value of the original conditional statement?

Contrapositive

Contrapositive of a Conditional Statement:

formed by negating both the hypothesis and the conclusion and then interchanging the resulting negations.

Put “not” into the hypothesis and the conclusion, then switch the order.

Example

Conditional: If 8 is an even number, then 8 is

divisible by 2.

Contrapositive:If 8 is not divisible by two, then 8

is not an even number.

HINT

For contrapositive, combine both converse and inverse.

The truth value of a contrapositive is ___________________ the original conditional statement.

The same as

Conditional, Converse, Inverse, and Contrapositive

For each statement, write the (a) converse, (b) inverse, and (c) the contrapositive. Give the truth value for each statement.

1. If a student is on the University of Arkansas football team, then he is called a Razorback . TRUEIf a student is called a Razorback, then he is on the University of Arkansas football team. FALSE

If a student is not on the University of Arkansas football team, then he is not called a Razorback. FALSE

If a student is not called a Razorback, then he is not on the University of Arkansas football team. TRUE

a

b

c

Conditional, Converse, Inverse, and ContrapositiveFor each statement, write the (a) converse, (b) inverse, and (c) the

contrapositive. Give the truth value for each statement.

2. If a person skateboards well, then they have a good sense of balance. TRUE

If a person has a good sense of balance, then they skateboard well. FALSE

If a person does not skateboard well, then they do not have a good sense of balance. FALSE

If a person does not have a good sense of balance, then they do not skateboard well. TRUE

a

b

c

Conditional, Converse, Inverse, and Contrapositive

For each statement, write the (a) converse, (b) inverse, and (c) the contrapositive. Give the truth value for each statement.

1. If it thunderstorms, then our pond overflows. TRUEIf our pond overflows, then it has thunderstormed. FALSE

If it does not thunderstorm, then our pond will not overflow. FALSE

If our pond does not overflow, then it has not thunderstormed. TRUE

a

b

c

Venn Diagrams

A Venn diagram is a drawing used to represent a set of numbers or conditions. Venn diagrams can be useful in explaining conditional statements.

Example

If you live in Bentonville, then you are a Tiger fan.

Hypothesis

Conculsion you live in Bentonville

you are a Tiger fan

Write the conditional statement from the Venn diagram.

Squares

Quadrilaterals

Equilateral

Triangles Triangles with 3 acute angles

Parallelogram with perpendicular diagonals

Rhombus

If an object is a square, then it is a quadrilateral.

If a triangle is equilateral, then it has three acute angles.

If a parallelogram has perpendicular diagonals, then it is a rhombus.