ImplicationsImplications

All prairie dogs are rodentsAll prairie dogs are rodents

If an animal is a prairie dog, then it is a If an animal is a prairie dog, then it is a rodent.rodent.

This IMPLIES thatThis IMPLIES that

EVERY prairie dog is a rodentEVERY prairie dog is a rodent

Hypothesis - If an animal is a prairie dogHypothesis - If an animal is a prairie dog

Conclusion - then it is a rodentConclusion - then it is a rodent

An animal is a prairie dog only if it is a An animal is a prairie dog only if it is a rodentrodent

ImplicationsImplicationsHypothesis – the “if” part of a Hypothesis – the “if” part of a conditional statementconditional statement

Conclusion – the “then” part of a Conclusion – the “then” part of a conditional statementconditional statement

Implication – a statement that cal be Implication – a statement that cal be written in the form p --written in the form p -- q (Read “p q (Read “p implies q” or “If p, then q”)implies q” or “If p, then q”)

Conditional – a statement that can be Conditional – a statement that can be written in the form p-written in the form p- q: q:

ImplicationsImplications

The New England states are east of the The New England states are east of the Mississippi RiverMississippi River

If a state is in New England,If a state is in New England,

Then it is east of the Mississippi.Then it is east of the Mississippi.

Counterexample –Counterexample –

Virginia is east of the Mississippi River, Virginia is east of the Mississippi River, but it is not a New England statebut it is not a New England state

ImplicationsImplicationsThis month is February This month is February

if today is Valentines Dayif today is Valentines Day

REWRITE THE IMPLICATIONREWRITE THE IMPLICATION

If today is Valentines DayIf today is Valentines Day

Then it’s the month of February Then it’s the month of February

TRUETRUE

ImplicationsImplicationsThis month is FebruaryThis month is February

if today is Valentines Dayif today is Valentines Day

WRITE THE CONVERSEWRITE THE CONVERSE

If it’s the month of FebruaryIf it’s the month of February

Then its Valentines DayThen its Valentines Day

FALSE FALSE

COUNTEREXAMPLE COUNTEREXAMPLE

It could be February 11thIt could be February 11th

ImplicationsImplicationsRectangles are parallelogramsRectangles are parallelograms

REWRITE THE IMPLICATIONREWRITE THE IMPLICATION

If the polygon is a rectangleIf the polygon is a rectangle

Then it is a parallelogramThen it is a parallelogram

TRUE TRUE

ImplicationsImplicationsRectangles are parallelogramsRectangles are parallelograms

WRITE THE CONVERSEWRITE THE CONVERSE

If the polygon is a parallelogramIf the polygon is a parallelogram

Then it is a rectangleThen it is a rectangle

FALSEFALSE

CounterexampleCounterexample

It could be a Rhombus It could be a Rhombus

ImplicationsImplicationsA number ending in 0 implies that it is A number ending in 0 implies that it is

divisible by 2.divisible by 2.

REWRITE THE IMPLICATIONREWRITE THE IMPLICATION

If a number ends in 0If a number ends in 0

Then the number is divisible by 2Then the number is divisible by 2

TRUE TRUE

ImplicationsImplicationsA number ending in 0 implies that it is A number ending in 0 implies that it is

divisible by 2.divisible by 2.

WRITE THE CONVERSEWRITE THE CONVERSE

If a number is divisible by 2If a number is divisible by 2

Then the number ends in 0Then the number ends in 0

FALSEFALSE

Counterexample: 24 is divisible by 2 but Counterexample: 24 is divisible by 2 but doesn’t end in 0 doesn’t end in 0

ImplicationsImplicationsCleo is a golden retriever Cleo is a golden retriever

only if Cleo is a dogonly if Cleo is a dog

REWRITE THE IMPLICATIONREWRITE THE IMPLICATION

If Cleo is a golden retrieverIf Cleo is a golden retriever

Then Cleo is a dogThen Cleo is a dog

TRUETRUE

ImplicationsImplicationsCleo is a golden retriever Cleo is a golden retriever

only if Cleo is a dogonly if Cleo is a dog

REWRITE THE CONVERSEREWRITE THE CONVERSE

If Cleo is a dogIf Cleo is a dog

Then Cleo is a golden retrieverThen Cleo is a golden retriever

False, counterexampleFalse, counterexample

It could be a German shepardIt could be a German shepard

ImplicationsImplicationsA Chord is a segment whose endpoints A Chord is a segment whose endpoints

are on a circleare on a circle

chordchord

ImplicationsImplicationsA Chord is a segment whose endpoints A Chord is a segment whose endpoints

are on a circleare on a circle

a.a.Is the diameter of a circle a chord?Is the diameter of a circle a chord?

b.b.Write an implication about chords and Write an implication about chords and diameters.diameters.

c.c.WRITING Write the converse of your WRITING Write the converse of your implication. Is it true? Explain implication. Is it true? Explain