Conditionals

LogicThe Conditional and Related Statements

Resources: Resources: HRW Geometry, Lesson 12.3HRW Geometry, Lesson 12.3

Introduction Instruction Examples Practice

Logic and Geometry are both about developing good arguments or proofs that something is true or false. The other two connectives that create compound statements in logic, the conditional statement, and the biconditional statement are often involved in arguments and proofs.

How can we tell if a conditional statement is true or false?


Please go back or choose a topic from above.

https://local.greenville.k12.sc.us/curric/6_12math/geom/docs/Logic,_Conditionals_FINAL.ppt

List of Instructional Pages

1. Two New Connectives2. The Conditional 3. The Conditional – Trut

h Table4. The Biconditional 5. The Biconditional – Tr

uth Table6. Other If…then Statem

ents7. The Converse

8. The Inverse9. The Contrapositive10.

Conditional/ Converse – Truth Tables

11.Conditional/ Inverse – Truth Tables

12.Conditional/ Contrapositive – Truth Tables

13. Summary





















We are ready to add the conditional and biconditional to our list of connectives.

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Negation:Negation: NOT

Conjunction:Conjunction: AND

Disjunction:Disjunction: OR

Conditional:Conditional: if…thenif…then

Biconditional:Biconditional: if and only ifif and only if

The Symbols:The Symbols:NOT ~

AND

OR

If…then

If and only if

The Connectives - The Connectives - Conditional and BiconditionalConditional and Biconditional






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The Connectives:The Connectives:The ConditionalThe Conditional

A conditionalconditional expresses the notion of if . . . then. We use an arrow, ,, to represent a conditional.

p : You will study hard.

s : You will get a good score in the exam.

p s : If you will study hard, then you will get a good score in the exam.

“If you will not study hard, then you will not get a good score in the exam.” would be written as ~p ~s.





There are three other if…then statements related to a conditional statement, p q. They are called: Converse: q pInverse: ~p ~qContrapositive: ~q ~p


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Not exactly the same thing in Geometry!

If..then statements related to conditionals

Do they all mean Do they all mean the same thing?the same thing?




Give the converse, inverse and contrapositive of the given conditional.

Conditional: p q “if p, then q”“If you are a Filipino, then you are Asian.”

Converse: q p “if q, then p” “If you are Asian, then you are a Filipino.”

Inverse: ~p ~q “if not p, then not q” “If you are not a Filipino, then you are not Asian.”

Contrapositive: ~q ~p “if not q, then not p” “If you are not Asian, then you are not a Filipino.”


Conditional: p q“If the figure is a square, then it has four sides.”

Converse: q p “If the figure has four sides, then it is a square.”

Inverse: ~p ~q “If the figure is not a square, then it doesn’t have four sides.”

Contrapositive: ~q ~p“If the figure doesn’t have four sides, then it is not a square”


Conditional:“If the animal has wings, then it is a bird.”

Converse: “If the animal is a bird, then it has wings.”

Inverse: “If the animal has no wings, then it is not a bird.”

Contrapositive: “If the animal is not a bird, then it does not have wings”


Conditional:“If the animal has feathers, then it is a bird.”

Converse: “If the animal is a bird, then it has feathers.”

Inverse: “If the animal has no feathers, then it is not a bird.”

Contrapositive: “If the animal is not a bird, then it does not have feathers.”

Give the converse, inverse and contrapositive of the given conditionals.

a : If today is Sunday, then it is a weekend day.

b : If the figure has three sides, then it is a triangle.

c : If Charlie is a basketball player, then Charlie is tall.

d : If you are a teenager, then you are 13 years old.

2. The car is washed but the ₱10 was not paid. The promise is not kept so the conditional is false.

4. The car is not washed and the ₱ 10 is not paid. The promise is not broken since the car was not washed, so the conditional is still true.

3. If the car was not washed, the payment does not have to be either true or false. Either way, the promise is kept, so the conditional is true.

1. The car is washed and the ₱ 10 is paid. The promise is kept so the conditional is true.


A conditional statement uses the words if…then. It is like making a promise. In logic, if the “promise” is broken, and not kept, the conditional is said to be false. Otherwise, it is true.Consider the statement: “If you wash my car, then I will pay you ₱10.” (p q) There are four situations possible.

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The ConditionalThe Conditional

p q p q

TT TT TT

TT FF FF

FF TT TT

FF FF TT




Let’s say p represents the statement “Marge lives in Cebu,” and q represents the statement “Marge lives in the Philippines.”

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p q is “If Marge lives in Cebu, then she lives in the Philippines.” The Converseq p “If Marge lives in the Philippines, then she lives in Cebu.”

Just because the conditional is true does not mean the converse

is true.

TRUETRUE

ConverseConverse FALSEFALSE





Let’s look at the inverse.


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p q is “If Marge lives in Cebu, then she lives in the Philippines.”

The Inverse

~p ~q “If Marge does not live in Cebu, then Marge does not live in the Philippines

Marge could still live in the Philippines and not be in Cebu.

Just because the conditional is true does not mean the

inverse is true.

Inverse

TRUETRUE

FALSEFALSE





Let’s look at the contrapositive.


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p q is “If Marge lives in Cebu, then she lives in the Philippines.”

The Contrapositive

~q ~p “If Marge does not live in the Philippines, then she does not live in Cebu.”

If Marge isn’t in the Philippines, she can’t be in Cebu.

If the conditional is true then the contrapositive is

also true.

ContrapositiveContrapositive

TRUETRUE

TRUETRUE




Slides after these are for optional study.


A biconditionalbiconditional expresses the notion of if and only if. Its symbol is a double arrow, ..

p : If a polygon has three If a polygon has three sides then it is a trianglesides then it is a triangle..

t : If a figure is a triangle If a figure is a triangle then it is a polygon with then it is a polygon with three sides.three sides.

p t : “A polygon has three sides if and only if it is a triangle.”


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The Connectives – the The Connectives – the BiconditionalBiconditional





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The BiconditionalThe Biconditional

pp qq p p q q q q p p (p (p q)q) ( (q q p) p)

(or p (or p q) q)

TT TT TT TT TT

TT FF FF TT FF

FF TT TT FF FF

FF FF TT TT TT

A biconditional (p t) is a more concise way to say (p t) (t p).

“If a polygon has If a polygon has three sides then it three sides then it is a triangleis a triangle” and “If a figure is a If a figure is a triangle then it is a triangle then it is a polygon with three polygon with three sidessides” are both true statements.

TT

A biconditional is true when both p q and q p are true.

TT





Let’s compare the truth tables for the conditional and the converse.


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Comparing Truth Tables Confirms Our Conjectures

pp qq p p q q

TT TT TT

TT FF FF

FF TT TT

FF FF TT

pp qq q q p p

TT TT TT

TT FF TT

FF TT FF

FF FF TT

The ConverseThe Converse


These two truth tables are not the same so the

statements are not logically equivalent.





Let’s compare the truth tables for the conditional and the inverse.


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Comparing Truth Tables Confirms Our Conjectures

pp qq p p q q

TT TT TT

TT FF FF

FF TT TT

FF FF TT

pp qq ~p~p ~q~q ~p ~p ~q ~q

TT TT FF FF TT

TT FF FF TT TT

FF TT TT FF FF

FF FF TT TT TT

The InverseThe Inverse


These two truth tables are not the same so the statements are not logically equivalent.





Let’s compare the truth tables for the conditional and the contrapositive.


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Comparing Truth Tables Comparing Truth Tables Confirms Our ConjecturesConfirms Our Conjectures

pp qq ~q~q ~p~p ~q ~q ~p ~p

TT TT FF FF TT

TT FF TT FF FF

FF TT FF TT TT

FF FF TT TT TT

The ContrapositiveThe Contrapositive

These two truth tables are the same so the

statements are logically equivalent.

pp qq p p q q

TT TT TT

TT FF FF

FF TT TT

FF FF TT






Let’s summarize the relationships:

Conditional and Contrapositive always has the same truth value.

Converse and Inverse always has the same truth value. This is page

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Summary

p q q p

p q ~p ~q

p q ~q ~p

ConverseConverse

InverseInverse

ContrapositiveContrapositive

q p ~p ~q








Example 1

Example 2

Example 3

Examples

1. Writing “if…then” statements

2. Writing the converse, inverse, or contrapositive of a conditional statement.

3. Recognizing the converse, inverse, contrapositive given a conditional statement.

IFIF

THENTHEN







PracticeGizmos: Conditional StatementBiconditional Statement

How can we tell if a conditional statement is true or false?



Example 1

All freshmen should report to the cafeteria.

Back to main example page

Rewrite each statement in if…then form. For example: “Every triangle is a polygon” becomes

““If a figure is a If a figure is a triangle, then the triangle, then the figure is a figure is a polygon.”polygon.”

If you are a freshman, then If you are a freshman, then you should report to the you should report to the cafeteria.cafeteria.

Reading horror stories at bedtime gives me nightmares.

If I read a horror story at If I read a horror story at bedtime, then I will have bedtime, then I will have nightmares.nightmares.

Driving too fast often results in accidents.

If you drive too fast, then If you drive too fast, then you are likely to have an you are likely to have an accident.accident.

Example 2ConverseConverse: If the football game was cancelled, then it must have rained all day Friday.


Write the converse, inverse, and contrapositive for the given conditional statement. Decide whether each is true or false and explain your reasoning.

““If it rains all day If it rains all day Friday, then the Friday, then the football game will football game will be cancelled.”be cancelled.”

False. The game could have been cancelled because of something else, like a bomb threat.

InverseInverse: If it did not rain all day Friday, then the football game was not cancelled.False. Just because it didn’t rain doesn’t mean the game couldn’t be cancelled for another reason.

ContrapositiveContrapositive: If the football game was not cancelled, then it did not rain all day Friday.True.

Example 3If I read the book, then I can do the homework.If I cannot do the homework, then I did not read the book.


For each statement, For each statement, name the relationship name the relationship (converse, inverse, (converse, inverse, contrapositive) of the contrapositive) of the second statement to the second statement to the first. State whether the first. State whether the second is always true second is always true (AT) or not always true (AT) or not always true (NAT) assuming p(NAT) assuming pq is q is true.true.

Contrapositive, ATContrapositive, AT

If it is Tuesday, I go to geometry.If I go to geometry, it is Tuesday

Converse, NATConverse, NAT

If it is snowing, then it is cold.If it isn’t snowing, then it isn’t cold.

Inverse, NATInverse, NAT

Class attendance will be down ifif the surf is up.If class attendance is down, then the surf is up.

Converse, NATConverse, NAT

Conditionals

Education

pleasego backor

sothe statements

study hard

logically

truth tables

good score

thenshe lives

statementmarge