mathematical reasoning form 4 mathematics

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Mathematical Reasoning Introducti Introducti on on All All Some Some Or Or and and If and If and only if only if IF IF All All Then Then - - Induction Induction Deduction Deduction

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A powerpoint presentation about Mathematical Reasoning that helps teacher to perform their Lesson. (The Document is a sharing from a course attended by me)

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Mathematical Reasoning

IntroductionIntroduction

AllAllSomeSome

OrOrandand

If and If and only ifonly if

IFIFAllAll

ThenThen

--

InductionInductionDeductionDeduction

(a) Is ‘3 + 2 = 5’ a statement ? Explain your answer. Yes, it is a statement

because it is true

(b) Is ‘3 + 2 = 6’ a statement ? Explain your answer.

Yes, it is a statement because it is false

(c) Is ‘3 + x = 5’ a statement ?

Explain your answer.No, it is not a statement

because we can’t determine whether it is true

 

Antecedence(cause)

Consequence(effect)If then

If x is a factor of 12 then x is a factor of 24

If m+5 = 12 then m = 7

If you work hard then you will score in the examination.

If n is an odd number then n+1 is an even numberIf x is a prime number then 2x is an even number

If you smoke then you will suffer fr lung cancer

All

Somepolygons have 5 sides.

False

True

All

Someboys are lazy.

All Some

False True

All triangles have 3 sides

All pentagons have 5 sides

Some triangles have 4 sides

All quadratic equations have two roots.

Some quadratic equations have two roots.

Some hexagons have 5 sides

Some prime numbers are odd numbers

All prime numbers are odd numbers

Some multiples of 3 are multiple of 6

All multiples of 3 are multiples of 6

True False

QUANTIFIER ‘‘ALL’’ AND ‘‘SOME’’

Based on the information given, construct a true statement using the quantifier ‘‘all’’ or ‘‘some’’

Object and propertyAnswer (true statement)

1 Acute angle ; less than 90o All acute angles are less than 90o

2 Negative number; smaller than zero

All negative numbers are smaller than zero

3 Triangle ; right-angled triangles Some triangles are right-angled triangles

Object and property Answer (true statement)

1 Rhombuses ; four equal sides All rhombuses have four equal sidesAll rhombuses have four equal sides

2 Odd number ; prime number Some odd number are prime numbersSome odd number are prime numbers

3 Factor of 6 ; factor of 3 Some factors of 6 are factor of 3Some factors of 6 are factor of 3

4 Isosceles triangle ; two equal sides

All isosceles triangles have two equal All isosceles triangles have two equal sidessides

5 Even number ; divisible by 10 Some even numbers are divisible by 10Some even numbers are divisible by 10

QUANTIFIER ‘‘ALL’’ AND ‘‘SOME’’

Based on the information given, construct a false statement using the quantifier ‘‘all’’ or ‘‘some’’

Object and propertyAnswer (true statement)

1 Null sets ; no elements Some null sets have no elements

2 Parallel line ; the same length

All parallel lines have the same length

3 Odd number ; perfect square

All odd number are perfect squares

Object and property Answer (true statement)

1 Orchid flower ; red in colour All orchid flower are red in colourAll orchid flower are red in colour

2 Human being ; heart Some human beings have heartSome human beings have heart

3 Quadratic equation ; 2 roots All quadratic equation have 2 rootsAll quadratic equation have 2 roots

4 2 digits number ; less than 100

Some 2 digits numbers are less than Some 2 digits numbers are less than 100100

5 Multiples of 8 ; can be exactly divided by 4

Some multiples of 8 can be exactly Some multiples of 8 can be exactly divided by 4divided by 4

Consequence(effect)

If then

Converse of a statement

Antecedence(cause)

you will score in the examination.

If Consequence(effect)

then Antecedence(cause)

If you want to score in the examination. you have to work hard

If you work hard then

then

x is a factor of 24

If x is a factor of 24 x is a factor of 12

If x is a factor of 12 then

then

If x > 5

The lines are parallel

their gradients are the same

If the gradient of two lines are the same

then

then

then x + 2 > 7

if x + 2 > 7If x > 5then

the hour and minute hands are in a stright line The time is 6 oclock then

then If the hour and minute hands are in a straight line The time is 6 o’clock

If two lines are parallel

2 + 3 = 5 3 2= 6and

or

False

True

Compound Statement

Statement I : – 4 > – 2

False

True

Combine the two statements below to form a new statement which is (a) true (b) false

Statement II : 3,4,5 is a Pythagorean triple

-4 > -2 and 3,4,5 is a Pythagorean triple

-4 > -2 or 3,4,5 is a Pythagorean triple

50 or 45 is a multiple of 10 11 00 1+0=11+0=1

6 is a factor of 16 or 24 00 11 0+1=10+1=1

A rectangle has 4 sides and a pentagon has 6 sides 11 00 1x0=01x0=0

7 is a factor of 49 and a prime number 11 11 1x1=11x1=1

12 + 22 = 32 and 32 + 42 = 52 00 11 0x1=00x1=0

2 is equal to 20 or (2 –1) –1 00 11 0+1=10+1=1

All even numbers are divisible by 2 or all odd numbers are divisible by 3

11 00 1+0=11+0=1

36 is a perfect square and a multiple of 4 11 11 1x1=11x1=1

80 is a perfect square or an even number 00 11 0+1=10+1=1

17 is a prime number and a factor of 34 11 11 1x1=11x1=1

1 m2 = 10 000 cm2 or 1 cm3 = 10 000 mm3 11 00 1+0=11+0=1

Ant is an insect and has 4 legs 11 00 1x0=01x0=0

if and only if

3x > 12

3x > 12

1.If

then

2.If then 3x > 12

1.If

then

then2.If

X > 4

X > 4

X > 4

Answ

er

Click

to sh

ow

(b) xy = 0 if and only if x = 0 or y = 0 

Implication I : If xy = 0 then x = 0 or y = 0 

Implication II : If x = 0 of y = 0 then xy = 0  

(a)    m < n if and only if 4m < 4n 

Implication I : If m < n then 4m < 4n  

Implication II : If 4m < 4n then m < n  

All regular pentagons have five equal sides

PQRST is a regular pentagon

PQRST has five equal sides.

SolveClick to show

Premise II :

Premise I :

Conclusion :

Premise II :

Conclusion :

All even number are divisible by 2

18 is an even number

18 is divisible by 2.

SolveClick to show

Premise II :

Premise I :

Conclusion :

Premise II :

Conclusion :

If x and y are odd number then the product of x and y is odd number5 and 7 are odd numbers

The product of 5 and 7 is odd number.

SolveClick to show

Premise II :

Premise I :

Conclusion :

Premise II :

Conclusion :

Premise I : All polygons have a sum of 180o for its exterior angles.

Conclusion : Octagon has a sum of 180o for its exterior angles

Octagon is a polygon.

SolveClick to show

Premise II :

Premise I : IF x is greater than zero, then x is a positive number.

Conclusion : 8 is a positive number

8 is greater than zero.

SolveClick to show

Premise II :

Premise I :

– 8 is a negative number

– 8 is less than 0

If x is a negative number then it is less than 0

SolveClick to show

Premise II :

Conclusion :

All negative numbers are less than 0

Premise I : If JKL is an equilateral triangle, then the value of its interior angles is 60o

Premise II : the value of JKL interior angles 60o

JKL is not an equilateral triangle

5 is a number that is a factor of 5

Number 5 is not a factor of 9

5 is a number that is not a factor of 9.

SolveClick to show

Conclusion :

Premise II : 11 is not divisible by 2

Conclusion : 11 is not divisible by 6

If x is a number that is divisible by 6 then x is divisible by 2

Premise 1 :

SolveClick to show

6 = (1)3 + 5

13 = (2)3 + 5

32 = (3)3 + 5

69 = (4)3 + 5 ……………………

+ 5 n = 1,2,3 …….. n = 1,2,3, 4 n = integer

SolveClick to show

n3

6 = (1)2 + 5

9 = (2)2 + 5

14 = (3)2 + 5

21 = (4)2 + 5 ……………………

+ 5

n = 1,2,3 …….. n = 1,2,3, 4 n = integer

12 = 2(2)3 – 4

50 = 2(3)3 – 4

124 = 2(4)3 – 4

246 = 2(5)3 – 4 ……………………

– 4

n = 2,3,4 …….. n =2,3, 4,5 n = integer

n2 2 n3

SolveClick to show

– 4 = 3 (1)2 – 7

5 = 3 (2)2 – 7

20 = 3(3)2 – 7

41 = 3(4)2 – 7

……………

- 7

n = 1,2,3 ……..

5 = 5 – 2(0)3

3 = 5 – 2(1)3

– 11 = 5 – 2(2)3

– 49 = 5 – 2(3)3

……………………

5

n = 0,1,2 ……..

141 = 2(3) + 5(3)3

328 = 2(4) + 5(4)3

635 = 2(5)+ 5(5)3

1092 = 2(6) + 5(6)3

……………………

2n

n = 3,4,5 ……..

n = 1,2,3,4 n = 0,1,2,3 n = 3,4,5 6.

n = integer n = integer n = integer

SolveClick to show

3n2 – 2 n3 +5 n3