Download “IF-THEN STATEMENTS” DAY 17 DEDUCTIVE REASONING: The

“IF-THEN STATEMENTS”
DAY 17
DEDUCTIVE REASONING: The process of using orderly statements to make logical conclusions.
IF-THEN STATEMENTS: CONDITIONAL STATEMENTS
_________________________________________________________________________
CONDITIONAL
CONVERSE (FLIPPED CONDITIONAL)
IF P THEN Q
IF Q THEN P
P is the Hypothesis; Q is the Conclusion
Q is the Hypothesis; P is the Conclusion
If a figure is a triangle, then it’s a polygon
If a figure is a polygon, then it’s a triangle
TRUE
FALSE
INVERSE
(CONDITIONAL with NOT)
If NOT P, then NOT Q.
If a figure it’s NOT a triangle,
Then it’s NOT a polygon
CONTRAPOSITIVE (FLIPPED CONDITIONAL WITH NOT)
If NOT Q, then NOT P.
If a figure it’s NOT a polygon,
then it’s NOT a triangle
FALSE
TRUE
CONDITIONAL and CONTRAPOSITIVE are LOGICALLY EQUIVALENT (either both True or both False)
CONVERSE and INVERSE are both LOGICALLY EQUIVALENT (either both True or both False)
_______________________________________________________________________________
GOAL:
ONLY USE GOOD DEFINITIONS TO PROVE SHORTCUTS…..
A GOOD DEFINTION WHEN THE CONDITIONAL AND CONVERSE ARE BOTH TRUE….
STATEMENT :
Perpendicular lines form right angles.
(A GOOD DEFINITION)
CONDITIONAL
CONVERSE
IF two lines are perpendicular,
IF two lines form right angles,
THEN they form right angles.
THEN the two lines are perpendicular
“TRUE”
“TRUE”
“IF AND ONLY IF” STATEMENTS: When Conditional and its Converse are both “TRUE”
BICONDITIONAL: Two lines are perpendicular “IF AND ONLY IF” they form right angles.
STATEMENT: Two lines are perpendicular if they form right angles (A GOOD DEFINITION)
STATEMENT:
A triangle is a polygon
(NOT A GOOD DEFINITON)
CONDITIONAL
IF a figure is a triangle, THEN it is a polygon.
TRUE
CONVERSE
IF a figure is a polygon, THEN it is a triangle.
FALSE
IF AND ONLY IF: NOT POSSIBLE
PAIR WORK: Express as a Conditional and a Converse Statement.
1. All equilateral triangles are isosceles
True or False.
(NOT A GOOD DEFINITION)
CONDITIONAL: If________________________________,
THEN__________________________
TRUE/FALSE
CONVERSE: If____________________________________,
THEN__________________________
TRUE/FALSE
IF AND ONLY IF STATEMENT: NOT POSSIBLE
2. Obtuse triangles have an obtuse angle.
(A GOOD DEFINITION)
CONDITIONAL: If________________________________,
THEN__________________________
TRUE / FALSE
CONVERSE: If____________________________________,
THEN__________________________
TRUE / FALSE
IF AND ONLY IF STATEMENT: A triangle is obtuse “IF AND ONLY IF” it has an obtuse angle.
C
x is positive.
32. x)4
lf p, then q.
Statement:
The square of an integer is odd.
33. An integer
is odd.
p, then not q.
If not
Inverse:
m are parallel.
Lines
intersect.
Lines
/DIAGRAMS
ar.d mIfdo
34.
a polygon.
it /isand
then
is MAKE
a triangle,
a not
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not atriangle.
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a figure is not a triangle,
is a right Ifangle.
3s.
inverse:
FalseLA
A polygon is regular.
A polygon is equilateral.
36.
DAY18
37. Alternate interior angles formed
Related
by lines
I and m and transversal
Summary
/ are congruent.
of
Lines
/
and m are parallel.
If-Then Statements
Given statement: If p, then q.
Contrapositive: lf not q, then not P.
38. a. Given: dnll oc; ADll BC
lf q, then p.
Converse:
Prove: 1A: /-C; LB: LD
If not p, then not q.
Inverse:
b. Tell what is given and what is to be proved in the converse of
converse.equivalent.
thelogically
contrapositive
a proof ofare
Thenitswrite
A statement
part (a). and
its inverse.
to an
into
and (b)or
to its(a)
converse
have equivalent
proved in parts
logically
not you
Combine iswhat
Ac.statement
if-and-only-if statement.
The relationships just summarized per
mit us to base conclusions on the contraposEULERDIAGRAMS
bvt not on
true if-then statement
itive of aConverse,
Contrapositive,
the converse
or inverse. For example, sup-
2-7
Inverse
true:
as an
this statement
accept
we the
pose
if-then statement and its conbetween
relationship
To
show
diagrams
to
use
circle
verse, it is helpful
All Olympic competitors,are athletes. (also called Venn diagrams
or
diagrams).
(If Euler
a person is an Olympic competitor, then
statement p, we draw a circle named p. If p is
To
represent
, person is an aathlete.)
that
true, we think of a point inside circle p. If p is false, we think of a
t
p is false.
p is true.
point outside circle p.
In the diagram at the left below, a point that lies inside circlep must also
lie inside circle q. In otherwords: If p,then q. Check to see that the middle
diagram represents the converse: If q, then p. Check the diagram at the right
92 /also.
Chapter
2
O
@
If 4, thenp.
\f p, then q.
p if and only if q.
Compare the following if-then statements.
Statement: lf p,lhen q.
Parallel Lines and Planes
/ 9l
Contrapositive: If not q, then not P.
You already know that the diagram at the right represents "lf p, then q."
The diagram also represents "If not Q, then not pi' because a point that
isn't inside circle q can't be inside circlep either. Since the statement and
its contrapositive are both true or else both false, they are called logically
equivalent. The following statements are logically equivalent.
True
statement: If a figure is a triangle, then it is a polygon.
Tiue contrapositive:
If
a hgure is not a polygon, then
it is not a triangle.
Since a statement and its contrapositive are logically equivalent, we may
prove a statement by proving its contrapositive. Sometimes that is easier.
There is one more conditional related to "If p, then q" that we will consider. A statement and its inuerse are not logically equivalent.
Statement: lf p, then
q.
Inverse: If not p, then not q.
True statement:
False inverse:
If
If
a figure is a triangle, then it is a polygon.
a figure is not a triangle, then it is not a polygon'
Summary of Related If-Then Statements
MOREEULERDIAGRAMS
Ex.IfcompetitorsareOlympiansthentheyareathletes
This statement is paired with four different statements below.
l.
Giuen:
lf p, then
q.
Olympic competitors
are athletes.
p
Ozzie is an Olympian.
Ozzie is an athlete.
Conclude: q
2. Giuen: lf p, then
All
q.
All
Olympic competitors
are athletes.
rlot q
Ned is not an athlete.
Conclude: not p
Ned is not an Olympic competitor.
3. Giuen: lf p, then q.
All
@
\
athletes
@
\
athletes
Olympic competitors
are athletes.
q
No conclusion follows.
Anne is an athlete.
Anne might be an Olympic
competitor or she might not
be.
4. Giuen: lf p,lhen
q.
Irot p
All
Olympic competitors
are athletes.
Nancy is not an Olympic
competitor.
No conclusion follows.
Nancy might be an athlete
or she might not be.
@i
Classroom Exercrses
1. State the contrapositive of each statement.
a.Ifx=3,thenx2+l:10.
.b. lfy(5,theny+6.
c. If a polygon is a triangle, then the sum of the measures of its angles is
180.
d. If you can't do it, then I can't do it.
2. State the converse of each statement in Exercise l.
3. State the inverse of each statement in Exercise 1.
4. A certain conditional is true. Must its converse be true? Must its inverse
be true? Must its contrapositive be true?
5. A certain conditional is false. Must its converse be false? Must its inverse
be false? Must its contrapositive be false?
Parallel Lines and Planes
/ 93
CW#18/HW#18
1.Given:Allsenatorsareatleast30yearsold.
a. Rewordthisstatementinif-thenform.
Conditional:Ifsomeoneisasenatorthenhe/sheisatleast30yrsold
Contrapositive:Ifsomeoneisyoungerthan30yrsoldthenhe/sheisnotasenator
b. MakeacirclediagramtoillustratetheConditionalstatement.
c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditional
statements?Ifnoconclusionispossible,sayno.
(Hint;OnlymakeconclusionspertheConditionalortheContrapositiveStatements)
1. JoseAvilais48yearsold.
________________________
2. RebeccaCastelloeisasenator ________________________
3. ConstanceBrownisnotasenator.
________________________
4. LingChenis29yearsold.
________________________
2.Given:Whenitisnotraining,Iamhappy
a. Rewordthisstatementinif-thenform.
b. Makeacirclediagramtoillustratethestatement.
c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditional
statements?Ifnoconclusionispossible,sayno.
1. Iamnothappy. __________________________
2. Itisnotraining. __________________________
3. Iamoverjoyed. __________________________
4. Itisraining.
_________________________
3.Given:Allmystudentslovegeometry
a.Rewordthisstatementinif-thenform.
b. Makeacirclediagramtoillustratethestatement.
c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditional
statements?Ifnoconclusionispossible,sayno.
1. Stuismystudent.
__________________________
2. Luislovesgeometry.
__________________________
3. Stellsisnotmystudent. __________________________
4. Georgedoesnotlovegeometry.
_________________________
INDIRECT REASONING /PROOFS
DAY 19
INDIRECT REASONING:
1. Uses the idea that if a CONDITIONAL is TRUE, then its CONTRAPOSITIVE is also TRUE.
CONDITIONAL: IF P THEN Q
CONTRAPOSITIVE: IF NOT Q THEN NOT P
2. Uses the CONTRAPOSITIVE as the INDIRECT REASONING
___________________________________________________________________________
USING INDIRECT REASONING
Explain how you would know if a driver applied the brakes.
STATEMENT:
A car leaves skid marks when it applies the brakes.
CONDITIONAL:
If a car leaves skid marks then it has applied the brakes
CONTRAPOSITIVE: If a car does not apply the brakes, then it will not leave skid marks.
INDIRECT REASONING:
If a car does not apply the brakes, then it will not leave skid marks.
Skid marks were left by the car. Therefore, the car must have applied the brakes.
____________________________________________________________________________
USING INDIRECT REASONING:
Explain why ice is forming on the sidewalk in front of Toni’s house.
STATEMENT:
Ice forms when it is 32F or below.
CONDITIONAL:
If ice forms then the temperature is 32F or below.
CONTRAPOSITIVE: If the temperature is more than 32F, then ice will not form on the sidewalk.
INDIRECT REASONING:
If the temperature is more than 32F, then ice will not form on the sidewalk..
Ice is forming on the sidewalk. Therefore, the temperature must be 32F or less.
PAIR WORK:
USING INDIRECT REASONING
Johnnie is too lazy to create flash cards. Explain how you know he isn’t going to get an A
STATEMENT:
Every student who gets an A in Geometry creates and uses flash cards.
CONDITIONAL:
If ____________________________ THEN _______________________
CONTRAPOSITIVE: IF ___________________________ THEN_________________________
INDIRECT REASONING:_______________________________________________________
______________________________________________________________________
INDIRECT PROOFS: PROVING BY CONTRADICTION
1. Assume temporarily that the conclusion is not true.
2. Reason logically until you reach a contradiction of a known fact
3. Therefore, the temporary assumptions must be false and what needs to be proven must be
true
____________________________________________________________________
Given (Hypothesis): n is an integer and n2 is even
Prove (Conclusion): n is even
Indirect Proof:
1. Assume temporarily that n is not even.
2. Then n is odd, and n X n = odd. This contradicts the given information that n2 is even.
3. Therefore, that n is not even must be false.
_________________________________________________________________
CW#19 / HW#19
What is the first sentence of an indirect proof of the statement shown?
1. Triangle ABC is equilateral.
______________________________
2. Doug is Canadian.
______________________________
3. a ≥ b
______________________________
4. Kim isn’t a violinist.
______________________________
5. Write an Indirect Proof
Given (Hypothesis): A triangle
Prove (Conclusion): There can be at most 1 right angle
a. Assume temporarily that ______________________________________________
b. Then _____________________________________________________________
___________________________________________________________________
c. Therefore __________________________________________________________
6.
Write an Indirect Proof
Given (Hypothesis): Fresh skid marks appear behind a green car at the scene
Prove (Conclude); The car must have applied the brakes.
a. Assume temporarily that ______________________________________________
b. Then _____________________________________________________________
___________________________________________________________________
c. Therefore __________________________________________________________
7.
Write an Indirect Proof
Given (Hypothesis): Ice is forming on the side walk.
Prove (Conclude); The temperature outside must be 32F or less.
a. Assume temporarily that ______________________________________________________
b. Then _____________________________________________________________________
_________________________________________________________________________
c. Therefore _________________________________________________________________
What conclusions, if any, can you make from each pair of statements?
8.
There are three types of drawbridges; bascule, lift and swing. This drawbridge doesn’t swing or
lift.
Conclusion: __________________________________________________________________
9.
If this were the day of the party, our friends would be home. No one is home.
Conclusion: __________________________________________________________________
10.
Every traffic controller in the world speaks English on the job. Sumiko does not speak
English
Conclusion: __________________________________________________________________
11.
If non-vertical lines are perpendicular, then the product of their slops is -1. The product
of the slopes of non-vertical lines is not -1.
Conclusion: ___________________________________________________________________