Implications Implications All prairie dogs are rodents All prairie dogs are rodents If an animal is a prairie dog, If an animal is a prairie dog, then it is a rodent. then it is a rodent. This IMPLIES that This IMPLIES that EVERY prairie dog is a rodent EVERY prairie dog is a rodent Hypothesis - If an animal is a Hypothesis - If an animal is a prairie dog prairie dog Conclusion - then it is a rodent Conclusion - then it is a rodent An animal is a prairie dog only if An animal is a prairie dog only if it is a rodent it is a rodent
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Implications All prairie dogs are rodents If an animal is a prairie dog, then it is a rodent. This IMPLIES that EVERY prairie dog is a rodent Hypothesis.
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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