Aim: Conditional, Converse & Inverse Course: Math Lit. Aim: What is a conditional statement? Its converse? Its inverse? Do Now: Construct a truth table

Aim: Conditional, Converse & Inverse Course: Math Lit.

Aim: What is a conditional statement? Its converse? Its inverse?

Do Now:Construct a truth table for the following compound statement:Alfie did not come last night and did not pick up his money.

p represents ‘Alfie did come last night.’

q represents ‘Alfie picked up his money.’

p q

T

T

F

F

T

F

T

F

T

F

F

T

F

F

T

T

F

F

T

F

~p ~q ~p ˄ ~q

The only time the compound statement is true is when both p and q are false.


Conditional StatementConditional Statement

• A conditional is a compound sentence formed by combining two simple sentences with the connective words “if . . ., then . . .”

• The “if” part of the conditional is called the antecedent, hypothesis or premise.

• The “then” part is called the conclusion or consequent.

If the glove fits, then you must acquit!

antecedent,hypothesis, premise

antecedent,hypothesis, premise

conclusion, consequent

conclusion, consequent

“if . . then” indicates a logical relationship – the consequent follows logically from the antecedent.


1. If you send in the proof of purchase, then they will send you a get well card.

2. If Mr. Mazes assigns homework, you’d better do it.

Rewrite each in if-then form and underline as above.

Identify the hypothesis and conclusion in the following by underlining the hypo once and the

conclusion twice:

1. “Vote for me and I’ll whip unemployment.”

2. For good health, exercise regularly.

Hidden conditionals

“If you vote for me, then I’ll whip unemployment.”

“If you exercise regularly, then you will have good health.”


If New Mexico is a state containing the word New, then New Mexico borders an ocean.

If 9 is an odd number, then 9 is a prime number.

Conditional and Truth ValuesConditional and Truth Values

Is this conditional statement true?

If the name of a state contains the word New, then the state borders an ocean.

To show that a conditional is false, you need to find only one example for which the hypothesis is true and the conclusion is false

New Mexico

F

F9 is not prime -


Find an example for each statement that shows it to be false.

1. If it is not a weekday, then it is Saturday.

2. If you live in a country that borders the United States, then you live in Canada.

3. If your last name is Vladimir, then you must be Russian.

Sunday

Mexico

SI Yankee Vladimir Hernandez is a Black Hispanic ball player


Conditional - Symbolically

antecedent if..p

consequent then..

q

What’s the truth?

“If I receive my check tomorrow, then I will pay you the $10 I owe you.”

Let p rep I receive my check tomorrow

Let q rep I will pay you the $10 I owe you.

p q p q

T T

T F

F T

F F

T

F

T

T

ConditionalTruth Table


Model Problems

r: It is Tuesday j: I am in school

Write symbolically:

1) If it is Tuesday, then I am in school

2) If it is not Tuesday, then I am not in school.

p: I am hungry q: I want a sandwich

Write sentence for

1) p q 2) q p

3) ~p q 4) ~q p

r j

~r ~j

1) If I am hungry, then I want a sandwich.

2) If I want a sandwich, then I am hungry.

3) If I am not hungry, then I don’t want a sandwich.

4) If I don’t want a sandwich, then I am not hungry.


Translations for Conditional

The if part of a conditional need not be stated first.Conditional translation Example

If p, then q. If you are 18, then you can vote.

q, if p. You can vote, if you are 18.

p, only if q You are 18 only if you can vote.

all p are q All 18-year-olds can vote.

hidden conditionals


Model Problems

Construct a truth table for the following statement. (p ˄ q) p

p q

T

T

F

F

T

F

T

F

F

T

F

F

T

T

T

T

p ˄ q (p ˄ q) p


Model Problems

If you do not itemize deductions on Schedule A and you have charitable contributions then complete the worksheet on page 14 and enter the allowable part on line 34b.

Understand the problemDevise a plan

Carry out the planLook back

d: You itemize deductions in Sched A.

c: You have charitable contributions.

w: You complete the worksheet on page 14.

b: You enter the allowable part on line 34b.

If not d and c, then (w and b).

(~d ˄ c) (w ˄ b)

Interpret the sentence.rephrase and represent symbolically


Model Problems

If you do not itemize deductions on Schedule A and you have charitable contributions then complete the worksheet on page 14 and enter the allowable part on line 34b.

If not d and c, then (w and b).

(~d ˄ c) (w ˄ b)

Is statement true if d, c, w, and b are all true?(~T ˄ T) (T ˄ T)

(F ˄ T) (T ˄ T)

F T

T


Converse and InverseConverse and Inverse

If a polygon is a quadrilateral, then it has four sides.

•If a polygon has four sides, then it is a quadrilateral.

•If a polygon is not a quadrilateral, then it does not have four sides.

Describe what happened to this conditional in each of the following:

Hypothesis and Conclusion switched places

Both Hypothesis and Conclusion were negated

ConverseConverse

InverseInverse


For the sentence that follows

a. Write a conditional & determine its truth value.

b. Write its converse and inverse

A right angle has a measure of 900.

a. If an angle is a right angle, then it’s measure is 900. T

b. Converse - If an angle’s measure is 900, then the angle is a right angle.

c. Inverse - If an angle is not a right angle, then it’s measure is not 900.


Write the converse and inverse for the following conditional and determine the truth value of all three sentences.

If a figure is a square, then it is a rectangle.

T

Converse - If a figure is a rectangle, then it is a square.

F

Inverse - If a figure is a not square, then it is not a rectangle. T


Summary of ConditionalsSummary of Conditionals

Statement Form

Conditional

Converse

Inverse

If , then

If , then

If not , then not

Conditional – If an angle is a straight angle, then its measure is 1800.

Converse – If an angle’s measure is 1800, then it is a straight angle.

Inverse – If an angle is not a straight angle, then its measure is not 1800.

- hypothesis - conclusion


Truth Table

p q ~p ~q pq ~p ~qq p

converse inverse

T

T

F

F

T

F

T

F

F

F

T

T

T

T

T

F

F

F

T

T

T

F

T

T

T

F

T

T

conditional


Assume the following statement is true:

p ~q If you obey the law, then you will not go to jail.

Model Problem

Write the converse and inverse and tell which are true.

converse: ~q p

If you do not go to jail, then you obey the law.

F

inverse: ~p q

If you do not obey the law, then you will go to jail.

F

p q

Documents

Aim: Conditional, Converse & Inverse Course: Math Lit. Aim: What is a conditional statement? Its converse? Its inverse? Do Now: Construct a truth table