Inductive and Deductive Reasoning (1).ppt

Modern mathematics

1. InductiveandDeductiveReasoning

2. InductiveReasoning
Thetypeofreasoningthatformsaconclusionbasedon
theexaminationofspecificexamples.Theconclusionformed
byusinginductivereasoningisoftencalledaconjecture,
sinceit mayormaynotbecorrect.
Inductivereasoningistheprocessofreachingageneral
conclusionbyexaminingspecificexamples.

3. ExampleofInductiveReasoningto
predicta number
3,6,9,12,15
Thepatternis
adifferenceof
3
1,3,6,10,15
Thepatternstarts
witha difference of
2then
continuouslyadd
one

4. ExampleofInductiveReasoningto
predicta number
5,10,15,20,25 2,5,10,17,26,
Thepatternis a difference
of5sothenextnumberis
30.
Thepatternstartswith
a difference of3then
continuously
add2.Nextnumberis37

5. UseInductiveReasoning
toMakeaConjecture
Considerthefollowingprocedure:Picka number.Multiply
thenumberby8,add6totheproduct,dividethesumby2,and
subtract3.Completetheaboveprocedureforseveral
differentnumbers.Useinductivereasoningtomakeaconjecture
abouttherelationshipbetweenthesizeoftheresultingnumber
andthesizeoftheoriginalnumber.
03

6. Solution
Originalnumber:5
Multiplyby8:8x5=40
Add6:40+6=46
Divideby2:46/2=23
Subtract3:23-3=20
Westartedwith5andfollowedtheproceduretoproduce20.
Startingwith6asouroriginalnumberproducesafinalresultof24.
Startingwith10producesafinalresultof40.Startingwith100produces
afinalresultof400.Ineachofthesecasestheresultingnumberisfour
timestheoriginalnumber

7. PENDULUM
Scientists
instance,
inductive
oftenuseinductivereasoning.For
usedGalileoGalilei(1564–1642)
reasoningtodiscoverthatthetime
requiredforapendulumtocompleteoneswing,
calledtheperiodofthependulum,dependsonthe
lengthofthependulum.Galileodidnothave
aclock,
pendulums
sohemeasuredtheperiodsof
in“heartbeats.”Thefollowingtable
showssomeresultsobtainedforpendulumsof
variouslengths.Forthesakeofconvenience,a
lengthof10incheshasbeendesignatedas1unit.

8. TABLE

9. Ifapendulumhasalengthof
49units,whatisitsperiod?
Ifthelengthofapendulum is
quadrupled,what happensto
itsperiod?
apendulumwithalengthof4unitshasaperiodthatis
twicethatofapendulumwithalengthof1unit.A
pendulumwithalengthof16unitshasaperiodthat
EXAMPLE
istwicethatofa
Itappears that
pendulumwithalengthof4units.
quadruplingthelengthofa
pendulumdoublesitsperiod.

10. EXAMPLE
Atsunamiisaseawaveproducedby
anunderwaterearthquake.Theheight
ofa tsunamiasitapproaches
landdependsonthevelocityofthe
tsunami.Usethetableattheleftand
inductivereasoningtoanswereachof
thefollowingquestions.

11. EXAMPLE
Whathappenstotheheightofatsunami
whenitsvelocityisdoubled?theheight
ofa tsunamiquadrupledif
itsspeedisdoubled.
Whatshouldbetheheightofatsunamiif
itsvelocityis30feetpersecond?100 ft

12. EXAMPLE
Conclusionsbasedon
inductivereasoningmay
beincorrect.Asan
illustration,considerthe
circlesshownbelow.
Foreachcircle,all
possiblelinesegments
havebeendrawnto
connecteachdotonthe
circlewithalltheother
dotsonthecircle.

13. EXAMPLE

14. EXAMPLE
Thereappearstobeapattern.Eachadditional
dotseemstodoublethenumberofregions.Guessthe
maximumnumberofregionsyouexpectfora
circlewithsixdots.Checkyourguessbycountingthe
maximumnumberofregionsformedbytheline
segmentsthatconnectsixdotsonalargecircle.
Yourdrawingwillshowthatforsixdots,themaximum
numberofregionsis31(seethefigureat
theleft),not32asyoumayhaveguessed.Withseven
dotsthemaximumnumberofregionsis57.Thisisa
goodexampletokeepinmind.
Justbecausea patternholdstrueforafew
cases,itdoesnotmeanthepatternwillcontinue.When
youuseinductivereasoning,youhavenoguarantee
thatyourconclusioniscorrect.

15. Counterexamples
Astatementisatruestatementprovidedthatitis
trueinallcases.Ifyoucan findonecase
forwhichastatementisnottrue,calleda
counterexample,thenthestatementisa false

16. ExampleofCounterexample
[x]>0
[0]>0
0 > 0 false
√x²=x
√(-3)²=-3false
x²>x
1²>1
1 > 1 false

17. ExampleofCounterexample
x
x
=1
x+3
=x+1
3
√x²+16=x+4

18. ExampleofCounterexample
0
0
=1
3+3
=3+1
3
Indeterminate 2=4
√3²+ 16= 3+ 4
Sqrt of 25=7
5 = 7

19. DeductiveReasoning

20. DeductiveReasoning
Deductivereasoningisdistinguishedfrominductivereasoningin
that itistheprocessofreachingaconclusionby
applyinggeneralprinciplesand procedures.

21. Example
Procedure:Pickanumber.Multiplythenumberby8,add 6
totheproduct,dividethesumby2,and subtract3.
Letnrepresenttheoriginalnumber. n
Multiplythenumberby8:
Add6totheproduct:
8n
8n+6
Dividethesumby2: 8n+6
=4n+3
2
4n+3-3=4nSubtract3:

22. Example
Procedure:Pickanumber.Multiplythenumberby6,add10
totheproduct,dividethesumby2,and subtract5.
Letnrepresenttheoriginalnumber.: n
Multiplythenumberby6:
Add10totheproduct:
6n
6n+10
Dividethesumby2: 6n+10
=3n+5
2
3n+5-5=3nSubtract5:

23. InductiveReasoning
vs.
DeductiveReasoning

24. InductiveReasoning
Duringthepast10years,atreehasproducedplums
everyotheryear.Lastyearthetreedidnotproduce
plums,sothisyearthetreewillproduceplums.
Thisargumentreachesaconclusionbasedon
specificexamples,soitisanexampleofinductive
reasoning

25. InductiveReasoning
IknowIwillwinajackpotonthisslotmachineinthenext10
tries,becauseithasnotpaidoutanymoneyduringthelast
45tries.
Thisargumentreachesaconclusionbasedonspecific
examples,soitisanexampleofinductivereasoning.

26. DeductiveReasoning
Allhomeimprovementscostmorethantheestimate.The
contractorestimatedthatmyhomeimprovementwillcost
$35,000.Thusmyhomeimprovementwillcostmorethan
$35,000.
Becausetheconclusionisaspecificcaseofageneral
assumption,thisargumentisanexampleofdeductive
reasoning.

27. DeductiveReasoning
AllJanetEvanovichnovelsareworthreading.
ThenovelTwelveSharpisaJanetEvanovich
novel.ThusTwelveSharpisworthreading.
Becausetheconclusionisaspecificcaseof
ageneralassumption,thisargumentisan
exampleofdeductivereasoning.

28. “Thereisalwaysareasonfor
everydecisionthatwemake.
Inductiveordeductive…doesn’t
matteraslongasitistheright
decision”
-GodofredoTrajanoTesorio