2. Who’sWho?
◆ Ali always speaks the truth
◆ Bala speaks the truth sometimes, and tells lies
sometimes
◆ Chin always tells lies
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The middle
person is Chin
The middle I am Bala.
person is Ali
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3. Prove It!
◆ The sum of three consecutive numbers, (n-1), n
and (n+1)is 3n
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4. Contoh Penaakulan Deduktif
Jika n ialah sebarang nombor bulat,
maka nombor sebelumnya ialah (n-1)
dan nombor selepasnya ialah (n+1)
Jadi, hasil tambah 3 nombor berturutan ialah
(n-1) + n + (n+1) = 3n
n n+1
n-1
6. The Nature ofDeductiveReasoning
◆ Begins with a general statement / hypothesis and
examines the possibilities to achieve a specific and
logical conclusion. (Bradford, A.,2017)
◆ Makes specificobservations/examplesfromgeneral
conclusions.
◆ Penaakulan deduktif ialah proses membuat
rumusan berdasarkan pernyataan umum
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7. The Nature ofDeductiveReasoning
◆ We infer conclusionsfrom knowninformation (called
premises) based on formal logic rules, and there is no
need to validate them by experiments1.
◆ If a general statement is assumed to be true and
another situation relates to the first assumption,then
the first statement must also hold true for the second
situation.
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1a classic approach of deductive reasoning
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8. The Nature ofDeductiveReasoning
◆ Formulateavalid,logical argumentto explain,
demonstrate or convince others that a solution to a
problem must be correct, or that a particular
conjecture is true or false.
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Haylock &Thangata (2007)
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9. Ada 12 keping duit syiling serupa.
Sekeping daripada duit syiling itu adalah
palsu.
Duit syiling palsu adalah ringan sedikit
daripada duit syiling benar.
Dengan hanya menggunakan sebuah
neraca mudah, bagaimanakah anda dapat
mencari syiling palsu dengan 3 kali
timbangan sahaja?
MASALAH DUIT PALSU
10. Charcoal is black.
• All blacks are charcoal.
• Not all blacks are charcoal.
Yang mana tidak logik?
Pernyataan
umum
Rumusan
tidak logikal
Rumusan
logikal
Penaakulan deduktif:
Menganalisis suatu pernyataan dan membuat
rumusan yang sah daripadanya secara deduktif.
11. STATEMENT
PERNYATAAN
What is a statement?
What is a truth value of statement ?
PERNYATAAN BUKAN PERNYATAAN
• Kerbau boleh terbang.
• Kota Bharu di Kelantan.
• Kota Kinabalu di
Sarawak.
• Apa khabar?
• Dia cantik.
• Ayat ini benar.
• 3 + 2 = 6
• 3 x 2 = 6
• 2 + a = 9
• 3 + 2
12. LATIHAN
Pernyataan ialah ayat yang sama ada benar atau
palsu, tetapi tidak boleh kedua-duanya.
AYAT Pernyataan atau tidak?
a) Tinggi Tashika ialah 182
cm.
b) Dorothy nampak cantik!
c) x – y = 5.
d) 4 + 2 < 5.
e) Di manakah kampung
awak?
• ya
• tidak
• tidak
• ya
• tidak
13. Truth Value
Nilai Kebenaran
Nilai kebenaran suatu pernyataan adalah sama ada
benar atau palsu.
PERNYATAAN NILAI KEBENARAN
a) Kerbau boleh terbang.
b) Kota Bharu di Kelantan.
c) Kota Kinabalu di
Sarawak.
• Palsu
• Benar
• Palsu
d) 3 + 2 = 6
e) 3 x 2 = 6
• Palsu
• Benar
f) 4 + 2 > 5 • Palsu
14. Who is who?
◆ Ali always speaks thetruth
◆ Bala speaks the truth sometimes, and tells lies
sometimes
◆ Chin always tells lies
Orang di tengah
ialah Chin
Saya ialah Bala.
Orang di tengah
ialah Ali.
Dua budak ini bukan Ali kerana
Ali akan kata dia ialah Ali.
Jadi, ini ialah Ali dan
dia cakap benar.
Jadi, ini
ialah Chin.
Ini ialah
Bala.
15. Ada 12 keping duit syiling serupa.
Sekeping daripada duit syiling itu adalah palsu.
Duit syiling palsu adalah ringan sedikit daripada duit syiling benar.
Dengan hanya menggunakan sebuah neraca mudah, bagaimanakah anda dapat
mencari syiling palsu dengan 3 kali timbangan sahaja?
Masalah Duit Palsu
Deduksi 1:
Jika tak seimbang,
duit palsu berada
di longgok sebelah
yang naik.
Deduksi 2:
Jika seimbang, duit
palsu berada
di longgok ketiga.
Pernyataan: Sebelah yang ada syiling palsu akan naik ke atas.
16. Euler Diagram
◆ Pronounced as ȯi·lər
◆ A diagram that consist of closed curves, used to
representrelationshipbetweenlogicalpropositionsor
sets;it is similar to a Venn diagram.
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17. Euler vs VennDiagram
◆ A Venndiagramshowsallpossiblelogicalrelationships
between the sets, while an Eulerdiagramonlyshows
existingrelationships.
◆ Click here to know more: https://goo.gl/wqeVkA
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VennDiagram EulerDiagram
18. Quantifiers
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i.e. ∀x(x2≥0) is translated as ‘the square of every
number is either zero or greater than zero.
Universalquantifier
,Ɐ Existential quantifier
Ɐx means all theobjectsxintheuniverse
For situation P(x), Ɐx P(x) means for all x,P(x)
holds. Or for every object x,P(x) is true.
Common terms used: All,Each,Every,None.
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19. Quantifiers
66
i.e. ∃x(x≥x2) is translated as ‘some numbers are
greater than or equivalent to its square.
Universal quantifier Existentialquantifier
,∃
∃x means thereisatleastanobjectx in the universe
For situation P(x), ∃x P(x) means there is at least an
object x that satisfies P(x).
Common terms used: Thereexist,Some,Few.
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23. Euler DiagramsToVerify LogicalArguments
◆ An argument is validif
⬦ thepremisesaretrueleadstheconclusiontobetrue.
⬦ thepremisesarefalseleadstheconclusiontobe
false
⬦ thepremisesarefalseleadstheconclusiontobetrue
◆ An argument is invalid if the premiseis true but denies
the conclusion (true and false; insufficient information
to prove the conclusion).
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24. Euler DiagramsToVerify LogicalArguments
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Ani: All goats are mammals.
Ben: All mammals have a back bone.
Chan: Therefore, a goat has a back bone.
Is Chan’s conclusion valid? Use an Euler
diagram to verify it.
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25. Euler DiagramsToVerify LogicalArguments
Ani: All goats (G) are mammals
(M).
Ben: All mammals have a back
bone (B).
Chan: Therefore, a goat has a back
bone.
Chan’s conclusion is valid
B
M
G
Goat x
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27. Euler DiagramsToVerify LogicalArguments
Ani: Some dinosaurs (D) eat meat (M).
Ben: I eat meat.
Chan: So, Ben is a dinosaur.
Chan’sconclusionisinvalidbecausethepremisesare
truebutit deniestheconclusion.Benmightormight
not be a dinosaur.
Ben can be in
either region
D
M
X X
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28. Euler DiagramsToVerify LogicalArguments
75
Ani: Some poodles bark too much.
Ben: Some dogs are poodles.
Chan: So, some dogs bark too much.
Is Chan’s conclusion valid? Use an Euler
diagram to verify it.
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29. Euler DiagramsToVerify LogicalArguments
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Ani: Some poodles bark too much.
Ben: Some dogs are poodles.
Chan: So, some dogs bark too much.
Chan’sconclusionisinvalidbecausethediagramagrees
to the premises but denies the conclusion.
Both diagram agrees to
the premises but there
are two conclusions
P Bark too
much
P Bark too
much
D D
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30. Semua pelajar lelaki pakai seluar.
• A pelajar lelaki.
• Jadi, A pakai seluar.
• B pakai seluar.
• Jadi, B pelajar lelaki.
Which conclusion is valid? why?
Pakai
seluar
Lelaki
31. Tiada pelajar lelaki pakai seluar.
• A pelajar lelaki.
•Jadi, A tidak
pakai seluar.
• B pakai seluar.
• Jadi, B bukan
pelajar lelaki.
Which conclusion is valid? why?
Pakai
seluar
Lelaki
• C bukan lelaki.
• Jadi, C pakai
seluar.
32. Sebilangan pelajar lelaki pakai seluar.
• A pelajar lelaki.
•Jadi, A pakai
seluar.
• B pakai seluar.
• Jadi, B pelajar
lelaki.
Which conclusion is valid? why?
Pakai
seluar
Lelaki
• C bukan lelaki.
• Jadi, C pakai
seluar.
36. TruthTabletoVerify TruthofStatements
a) 5 is an even number ˅ 8 is an odd number
.
b) 5 is an even number ˅ 8 is an even number .
c) 5 is an odd number ˅ 8 is an even number .
p q p˅q
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37. Typeof Statements: ConditionalStatement
◆ If p,then q is written as p →q.
◆ i.e.
◆ If this is a triangle, then it has three straight
sides.
◆ B is a triangle →B has three straight sides
Premise / antecedent conclusion / consequent
p →q
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39. TypeofStatements
Types of Statement Symbol Meaning
Conditional (bersyarat) p → q If p, then q
Inverse (songsangan) ~p → ~q If not p, then not q
Converse (akas) q → p If q, then p
Contrapositive
(kontrapositif)
~q → ~p If not q, then not p
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40. Build Statements
Types of
Statement
Symbol Statement
Inverse ~p → ~q If x is not an even number, then x is not a
multiple of 2
Converse q → p If x is a multiple of 2, then x is an even
number
Contrapositive ~q → ~p If x is not a multiple of 2, then x is not an
even number
◆ If x is an even number
,then x is a multiple of 2.
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41. True or false?
Jika hari hujan, maka tanah akan jadi basah.
Hari hujan.
Jadi tanah basah.
Hari tidak hujan.
Jadi tanah tidak basah.
Tanah tidak basah.
Jadi, hari tidak hujan.
Tanah basah.
Jadi, hari hujan.
Inverse →
Converse →
Contrapositive →
p → q
~p → ~q
q → p
~q → ~p
42. p → q
Jika p, maka q.
Perempuan pandai memasak.
• Jika anda perempuan, maka anda pandai memasak.
Harga barang akan naik bila harga petrol naik.
• Jika harga petrol naik, maka harga barang akan naik.
Segitiga ada 3 sisi.
• Jika ini bentuk segitiga, maka ia ada 3 sisi.
Pernyataan p → q dalam bentuk ayat biasa:
44. Euler DiagramsToDetermineThe Truth
p : Polygon X is a square.
q : Polygon X has four sides of equal
length.
True because all squares have four sides
of equal length
If a polygon is a square, then it has four sides of equal
length.
Determine its truth.
p
85
q
X
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45. Euler DiagramsToDetermineThe Truth
p : Polygon X is a square.
q : Polygon X has four sides of equal
length.
Converse: If a polygon has four sides of
equal length, then it is a square.
False because there are polygons with four sides of
equal length that is not a square
If a polygon is a square, then it has four sides of equal
length.
Determine the truth of its converse.
p
86
q
X
X
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46. Euler DiagramsToDetermineThe Truth
p : Polygon X is a square.
q : Polygon X has four sides of equal
length.
Inverse: If a polygon is not a square, then it
does not have four sides of equal
length
False because there are non-square polygons that do
not have four sides of equal length
If a polygon is a square, then it has four sides of equal
length.
Determine the truth of its inverse.
p
X 87
q
X
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47. Euler DiagramsToDetermineThe Truth
p : Polygon X is a square.
q : Polygon X has four sides of equal
length.
Inverse: If a polygon does not have four sides
of equal length, then it is not a
square.
True because a polygon that does not have four
sides of equal length is outside the region of a
If a polygon is a square, then it has four sides of equal
length.
Determine the truth of its contrapositive.
p
X 88
q
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48. Analyse ConditionalandBiconditional
Statements
◆ A biconditional statement is a combinationof a
conditional statementanditsconversewritten in the
‘if and only if’ form.
◆ A biconditional is true if and only if both the
conditional statements are true.
◆ Biconditionals are represented by the symbol ⇔ or ↔
◆ i.e. A polygon is a triangle if and only if it has 3 sides
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Ifpolygonisatriangle,thenit has3 Ifpolygonhas3sides,thenit isa
sides triangle
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49. Analyse ConditionalandBiconditional
Statements
Apolygonisatriangleif andonlyif it has3sides
◆ Is this a biconditional statement?
⬦ Yes,becauseit hasthephrase‘if andonlyif’
◆ Is the statement true? Analyse the following.
⬦ Conditional statement: Ifpolygonisatriangle,then
it has3sides(true)
⬦ Converse statement: Ifpolygonhas3sides,thenit
isatriangle(true)
⬦ Therefore,thebiconditional statementistrue. 90
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50. Analyse ConditionalandBiconditional
Statements
x=
3if andonlyif x2=
9
◆ Is this a biconditional statement?
⬦ Yes,becauseit hasthephrase‘if andonlyif’
◆ Is the statement true? Analyse the following.
⬦ Conditional statement: if x=3thenx2=9(true)
⬦ Converse statement:if x2=
9thenx=
3(howabout
x=
-3?False)
⬦ Therefore,thebiconditional statementisfalse
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Counter
example
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