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TMA6123 Question Bank (Topic 1 -5)

Total questions: 95

Worksheet time: 3hrs 46mins

Name
Class
Date
1.

What is the name of the following format of logical argument?

If A, then B.

A.

Therefore B.

a)

Modus Ponens

b)

Modus Tollens

c)

Law of Syllogism

d)

Disjunctive Syllogism

2.

Let M(x) be the statement " x is Discrete Mathematic's students" and x  is the set of all people.

 Express the "Nobody is Discrete Mathematic's students" in symbolic form. (NOTE: ~ is negation)

a)

∃x M(x)\exists_x\ M\left(x\right)  

b)

∼∀x M(x)\sim\forall_{x\ }M\left(x\right)  

c)

∀x∼M(x)\forall_x\sim M\left(x\right)  

d)

∼∃x M(x)\sim\exists_{x\ }M\left(x\right)  

3.

Let P(x) be the predicate "x likes reading" and Q(x) be the predicate “x can speak French”. The domain for both predicates are lecturers in MMU . Use quantifiers and logical connectives to express each of the following statement. (NOTE: ~ is negation)


“Someone can speak French.”

a)

b)

c)

d)

4.

Let P(x) be the predicate "x likes reading" and Q(x) be the predicate “x can speak French”. The domain for both predicates are lecturers in PBU . Use quantifiers and logical connectives to express each of the following statement.


“Ali cannot speak French.”

a)
b)
c)
d)
5.

Choose the correct answer for the following truth table:

a)

T T T F

b)

F T T F

c)

T T T T

d)

T T F F

6.

Choose the correct answer for the following truth table:

a)

T T T F

b)

T F T F

c)

T T T T

d)

T T F F

7.

P : I am young

Q : I am happy


Write “I am old but happy” in symbolic form.

a)
b)
c)
d)
8.
The image in an example of a _____?
a)
Logical Equivalent Table
b)
Related Conditional Table
c)
Truth Table
d)
Conditional Table
9.

If Bill wears a green shirt, then he will pass the test. Bill wore a green shirt.


What can you conclude?

a)

Bill wore a green shirt.

b)

Bill passed the test.

c)

Nothing

10.

Find the truth value of given statement : 4+3=9

a)

False

b)

True

11.

Which of the following statement is a proposition?

a)

Get me a glass of milkshake

b)

God bless you!

c)

What is the time now?

d)

The only odd prime number is 2

12.

Let M(x) = x loves Mathematics,

where x consists of all students.


Translate ∀xM(x) into English sentence

a)

All students love Mathematics.

b)

Some students love Mathematics.

c)

There exists students love Mathematics.

d)

There is a student loves Mathematics.

13.

Let M(x) = x loves Mathematics,

where x consists of all students.


Translate ∃x¬M(x) into English sentence

a)

There is a student who is does not love Mathematics.

b)

There is a student who is loves Mathematics.

c)

All students does not love Mathematics.

d)

All students love Mathematics.

14.

p→(p∨q)p\rightarrow\left(p\vee q\right)  is a tautology

a)

True. It is a tautology

b)

No. It is contradiction

c)

none of the above

15.

If A={a,b,c} theen R={(b,c)}, then R is

a)

Reflexive only

b)

Symmetric only

c)

Transitive only

d)

equivalence relation

16.

Let A={1,2,3}, and R is {(1,1),(1,2),(1,3), (2,2), (2,3), (3,3)}, then R is

a)

reflexive but not symmetric

b)

reflexive but not transitive

c)

symmetric and transitive

d)

neither symmetric nor transitive

17.

Let N be a set of natural numbers and relation R on N be defined as

R = { (a,b):a,b are natural numbers and a + b =10 }

R is

a)

reflexive

b)

symmetric

c)

transative

d)

none of these

18.

Let set A = { 1, 2, 3, 4 } and R is a relation on A such that

R = { (1, 1), (2,2), (3,3) , (4,4) } then

a)

R is not reflexive relation

b)

R is not symmetric relation

c)

R is not transitive relation

d)

R is an equivalence relation

19.

R is a relation on set of natural numbers such that

R = { (a,b) : a + 4b = 10}

a)

R is a reflexive relation.

b)

R is a symmetric relation.

c)

R is a transitive relation.

d)

All of the above.

20.

Let R be a relation in the set {1,2,3,4} given by R={(1,2), (2,2), (1,1), (4,4), (1,3), (3 ,3), (3,2)} Then R is

a)

a) R is Reflexive and symmetric but not transitive

b)

a) R is Reflexive and transitive but not symmetric

c)

a) R is symmetric and transitive but not reflexive

d)

a) R is an equivalence relation

21.

Let R be the relation in the set N given by R = {(a ,b) : a = b – 2 , b > 6}, Then 

a)

 (2 , 4) ∈\in   R

b)

(3 , 8) undefined  R

c)

(6 , 8) undefined  R

d)

(8 , 7) undefined  R

22.

Given a set A = {1 , 2, 3} and a relation R = {(1 , 2), (2 ,1)} Then R will be

a)

Reflexive if (1,1) is added

b)

Symmetric if (2, 3) is added

c)

Transitive if (1 , 1) is added

d)

Symmetric if (2 ,3) is added

23.

R is a relation on set N of natural numbers Defined by

R = { (a,b) : b = a + 1 , a , b < 7 }.

a)

R is a reflexive relation.

b)

R is a symmetric relation.

c)

R is a transitive relation

d)

None of the above

24.

Let N be a set of natural numbers and relation R on N be defined as

R = { (a,b):a,b are natural numbers and a + b =10 }

R is

a)

reflexive

b)

symmetric

c)

transative

d)

none of these

25.

R is a relation on set of natural numbers such that

R = { (a,b) : a + 4b = 10}

a)

R is a reflexive relation.

b)

R is a symmetric relation.

c)

R is a transitive relation.

d)

All of the above.

26.

Let N be a set of natural numbers and relation R on N be defined as

R = { (a,b):a,b are natural numbers and a + b =10 }

R is

a)

reflexive

b)

symmetric

c)

transative

d)

none of these

27.

What value will be returned by the call recur(4)?

a)

1

b)

2

c)

3

d)

5

e)

8

28.
According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove. What is the first?
a)
The statement is true for n = 1.
b)
The statement is true for n = k.
c)
The statement is true for n = k+1.
29.
1 + 3 + 5 + 7 + . . . + (2n − 1) = n2
On the basis of this assumption,
[The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k2]
What must we show?
a)
The statement is true for n = 1:
2x1 − 1 = 12
b)
The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k2
c)
The statement is true for n = k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)2
30.
Let S(n) = 2n − 1. Evaluate: 
a)  S(k)
b)  S(k + 1)
a)
a)  S(k)  = 2k − 1
b)  S(k + 1) = 2n + 1
b)
a)  S(k)  = 2k + 1
b)  S(k + 1) = 2k + 1
c)
a)  S(k)  = 2k − 1
b)  S(k + 1) = 2k + 1
31.

The Cartesian Product B x A is equal to the Cartesian product A x B.

a)

TRUE

b)

FALSE

32.

Which of the following is true about the relation R defined in the set  {1,2,3,4,5} as R = {(a, b) : b = a + 1}?

a)

Symmetric

b)

Reflexive

c)

Transitive

d)

None of these

33.

Given the following A = {(1,1), (1,2), (1,3),(2,2),(2,1),(2,3),(3,3),(3,1),(3,2)}.

Find the in-degree and out-degree for each of the vertices and state the property of A.

a)

Reflexive, Symmetric, Transitive

b)

Transitive

c)

Reflexive, Symmetric

34.

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

a)

5

b)

10

c)

15

d)

20

35.

In how many ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

a)

120

b)

720

c)

4320

d)

2160

e)

None of these

36.

How many words with seven letters are there that start with a vowel and end with an A? (Note that they do not have to be real words and letters can be repeated.)

a)

45087902

b)

64387659

c)

12765800

d)

59406880

37.

If 4 Maths books are selected from 6 different Maths books and 3 English books are chosen from 5 different English books, how many ways can the seven books be arranged on a shelf?

a)

None of these ways

b)

750000 ways

c)

756000 ways

d)

480000 ways

38.

n ≥ 1

13 + 23 + 33 + · · · + n3 = [n(n + 1)/2]2

What would be the appropriate first step in proving the above mathematical induction?

a)

put n = 1

b)

put n = -1

c)

put n = 0

d)

none of these

39.

p: The bear is huge.

       q: The bear is a wild animal.

       r: The bear can eat human meat.

 Translate the following symbolic form into proper English.

( ∼ r ∧ ∼ p) → ∼ q

a)

If the bear can not eat human meat and the bear is not huge then, the bear is a wild animal.

b)

The bear is huge and the bear is a wild animal.

c)

If the bear can eat human meat and the bear is not huge then, the bear is a wild animal.

d)

If the bear can not eat human meat and the bear is not huge then, the bear is not a wild animal.

40.

Determine the missing term in the sequence: 4, 12, __, 108, 324, 972

a)

24

b)

38

c)

36

d)

72

41.

Find the missing number in the sequence: 5, 10, 15, ___, 25, 30.

a)

20

b)

35

c)

18

d)

24

42.

Buktikan bahwa 2n + 1 adalah bilangan ganjil untuk setiap n bilangan bulat non-negatif, menggunakan induksi matematika.

a)

2n + 1 adalah bilangan ganjil untuk setiap n bilangan bulat non-negatif.

b)

2n + 1 adalah bilangan prima untuk setiap n bilangan bulat non-negatif.

c)

2n + 1 adalah bilangan negatif untuk setiap n bilangan bulat non-negatif.

d)

2n + 1 adalah bilangan genap untuk setiap n bilangan bulat non-negatif.

43.

Buktikan bahwa 1 + 2 + 3 + ... + n = n(n + 1)/2 untuk setiap n bilangan bulat positif, menggunakan induksi matematika.

a)

n(n + 1)/2

b)

n-1

c)

n^2

d)

2n

44.

Buktikan bahwa 3^n - 1 adalah kelipatan dari 2 untuk setiap n bilangan bulat positif, menggunakan induksi matematika.

a)

3^n - 1 adalah kelipatan dari 2 untuk setiap n bilangan bulat positif.

b)

3^n - 1 adalah kelipatan dari 3 untuk setiap n bilangan bulat positif.

c)

3^n - 1 adalah kelipatan dari 5 untuk setiap n bilangan bulat positif.

d)

3^n - 1 adalah bilangan prima untuk setiap n bilangan bulat positif.

45.

Buktikan bahwa 2^n > n^2 untuk setiap n bilangan bulat positif, menggunakan induksi matematika.

a)

2^n > n^2 untuk setiap n bilangan bulat positif

b)

2^n < n^2 untuk setiap n bilangan bulat negatif

c)

2^n < n^2 untuk setiap n bilangan bulat positif

d)

2^n = n^2 untuk setiap n bilangan bulat positif

46.

What do you call to a mathematical statement or property considered to be self-evidently true, but yet cannot be proven?

a)

Axioms

b)

Fractions

c)

Modulus

d)

Integers

47.

A graph is denoted by a set of ordered pairs G = (V, E) as described below EXCEPT


Tatatanda graf merupakan set pasangan tertib iaitu G = (V, E) dengan keadaan berikut KECUALI

a)

V is the set of dots or vertices.

V ialah set bintik atau bucu.

b)

E is the set of edges or lines linking each pair of vertices.

E ialah set tepi atau garis yang menghubungkan sepasang bucu.

c)

Σd(v) = 2E ; v ∈ V

d)

V = E

48.

Which of the following statement is true about Simple Graph?

Manakah antara pernyataan di bawah adalah benar mengenai Graf Mudah?

a)

A simple graph has no loops and no multiple edges.

Graf mudah ialah graf yang tidak mengandungi gelung atau berbilang tepi.

b)

A simple graph has a loop but no multiple edges.

Graf mudah ialah graf yang mengandungi satu gelung tetapi tidak berbilang tepi.

c)

A simple graph has no loop but with multiple edges.

Graf mudah ialah graf yang tidak mengandungi satu gelung tetapi mempunyai berbilang tepi.

d)

The sum of degrees of the graph is equal to the

number of edges.

Bilangan darjah adalah sama dengan bilangan tepi.

49.

Choose all the simple graph below.

Pilih semua graf mudah di bawah.

a)
b)
c)
d)
e)
50.

Choose all the correct information for the simple graph given.

Pilih semua informasi yang benar bagi graf mudah yang diberi.

a)

V = {A, B, C, E, F, G}

b)

V = {A, B, C, D, E, F, G}

c)

E = {A, B, C, D, E. F. G}

d)

E = {AB, AC, BC, BD, BE, CD, CF, DE, DF, EF, EG, FG}

e)

n(E) = 12

51.

Choose all the correct information about the simple graph given.

Pilih semua informasi yang benar bagi graf mudah yang diberi.

a)

n(V) = 4

b)

n(V) = 5

c)

n(E) = 4

d)

n(E) = n(V)

e)

Sum of degree = 10

Bilangan darjah = 10

52.

Choose all the correct information about the graph given.

Pilih semua informasi yang benar bagi graf yang diberi.

a)

n(V) = 4

b)

n(E) = 3

c)

Sum of degree = 6

Bilangan darjah = 6

d)

d(v3)=2d\left(v_3\right)=2

e)

d(v3)+d(v6)=3d\left(v_3\right)+d\left(v_6\right)=3

53.

Choose all the correct information for the graph given.

Pilih semua informasi yang benar bagi graf yang diberi.

a)

This is a simple graph.

Ini ialah graf mudah.

b)

n(V) = 4

c)

n(E) = 10

d)

Sum of degree = 17

Bilangan darjah = 17

e)

Sum of degree = 20

Bilangan darjah = 20

54.

Choose all the correct information about the graph given.

Pilih semua informasi yang benar bagi graf yang diberi.

a)

V={v1, v2, v3, v4, v5}V=\left\{v_1,\ v_2,\ v_3,\ v_4,\ v_5\right\}

b)

E={1, 2, 3, 4, 5, 6, 7, 8}E=\left\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8\right\}

c)

Sum of degree = 16
Bilangan darjah = 16

d)

d(v3)=4d\left(v_3\right)=4

55.

Choose all the options with the degree given below that can form a graph.

Pilih semua opsyen dengan bilangan darjah yang diberi di bawah yang dapat membentukkan satu graf.

a)

3, 2, 1, 2

b)

2, 1, 1, 3

c)

3, 1, 2, 1, 1

d)

1, 1, 1, 3, 2

56.

What is the degree of Vertex D?

a)

1

b)

2

c)

3

d)

4

e)

5

57.

Check all of the paths from A to G.

a)

ADEEBFG

b)

ABFHG

c)

ADCBFG

d)

ABFG

e)

ABCBFG

58.

Check all of the vertices adjacent to Vertex B

a)

A

b)

C

c)

D

d)

E

e)

F

59.

What is the degree of vertex E?

a)

1

b)

2

c)

3

d)

4

e)

5

60.

Which of the following is a loop?

a)

BC

b)

EE

c)

AB

d)

BF

e)

none

61.

Which of the following is not a path from D to A/

a)

DCA

b)

DEDCCA

c)

DCCA

d)

DECDA

e)

All of the above are paths from D to A

62.

Choose all vertices that are adjacent to vertex A

a)

B

b)

C

c)

D

d)

E

63.

Which is an example of a disconnected graph?

a)
b)
c)
d)
e)

None are disconnected graphs

64.

What is the number of degree for the graph given?

a)

7

b)

6

c)

14

d)

12

65.

Which of the following is the subgraph of Graph G?

a)
b)
c)
d)
66.

Determine the number of vertices and sum of degrees.

a)

8,16

b)

8,8

c)

6,4

d)

8,13

67.

Determine n(E).

a)

7

b)

12

c)

14

d)

6

68.

Determine n(V).

a)

4

b)

8

c)

5

d)

10

69.

A loop is counted as _____ edge(s) but _____ degree(s).

a)

one, two

b)

one, one

c)

two, two

d)

two, one

70.

State the number of vertices.

a)

14

b)

3.5

c)

7

d)

5

71.

State Σd(V) .

a)

5

b)

7

c)

10

d)

14

72.

Which set represents the vertices of the graph?

a)

{1,2}, {2,3}. {2,4}, {4,5}, {4,6}

b)

{1, 2, 3, 4, 5, 6,}

c)

{1, 2, 3, 4, 5, 6, 7}

d)

{1,2}, {2,3}. {2,4}, {4,5}, {4,6}, {6,7}

73.

What is the degree of vertex 4?

a)

3

b)

4

c)

5

d)

7

74.

Which two vertices are adjacent vertices?

a)

5 is adjacent to 6

b)

3 is adjacent to 6

c)

4 is adjacent to 1

d)

3 is adjacent to 2

75.

Which vocabulary term describes the graph?

a)

circuit

b)

path

c)

complete graph

d)

polygon

76.

The graph represents the following vocabulary term.

a)

Cycle

b)

Path

c)

Complete Graph

d)

Complete Bipartite Graph

77.

Which vertices are adjacent to E?

a)

B and C

b)

B, C, D, and A

c)

B

d)

A, B, C, D, F, G

78.

Which of the following are isolated vertices?

a)

F

b)

G

c)

Graphs are fun.

d)

F and G

79.

Tina, Jessie, John, Bill, and Andy are all members of the social networking website Facebook. The site allows members to be “friends” with each other. It turns out that Bill and John are friends, as are Tina and Andy. Jessie is friends with everyone. Who is "E" ?

(a)  

80.

By looking at the pattern of the numbers in the table, determine the number of edges for a complete graph to be drawn from 9 nodes given.

a)

28

b)

36

c)

45

d)

55

81.

State the number of edges.

a)

18

b)

7

c)

9

d)

6

82.

The diagram shows a simple graph. State the sum of degrees of the graph.

(a)  

83.

The diagram shows a graph with a loop and multiple edges.

Which of the following are true?

a)

n(V) =4, n(E) =6

b)

n(V)=4,n(E)=8

c)

n(V)=6,n(E)=4

d)

n(V)=8,neE)=4

84.

The table shows the sets of edges of four graphs where each graph has four vertices.

Which graph is a simple graph?

a)

P

b)

Q

c)

R

d)

S

85.

Which of the following has a sum of degrees such that a graph can be drawn?

a)

3, 3, 1, 1

b)

3, 2, 2, 2

c)

2, 3, 3, 1

d)

2, 3, 1, 1

86.

Which describes the edges of the graph?

a)

{A,B}, {A,C}, {A, E}, {B,C}, {B,E}

b)

A, B, C, D, E

c)

{A,B}, {A,C}, {A, E}, {B,C}, {B,E}, {C,D}

d)

{A,B}, {A,C}, {A, E}

87.
Algebraic structures is any formal mathematical systems consisting of a set of ___ and ___ on those objects. 
a)
objects, operations
b)
objects, multiplication
c)
subjects, operations
d)
subjects, multiplication
88.
An algebraic structure is a semi group (S, *) if it is ____.
a)
closure
b)
commutative
c)
associative
d)
distributive
89.
Which of the following is TRUE?
a)
Associative property: (a * b) * c = a * (b * c)
b)
Commutative property: (a * b) * c = a * (b * c)
c)
Distributive property: (a * b) * c = a * (b * c)
d)
Associative property: a*b=b*a
90.
A monoid is a semi group that has ____.
a)
an inverse.
b)
an identity.
91.
An algebraic structure (G, *) is a group if it has the following properties:
a)
it is associative, has an identity, has an inverse
b)
it is associative, has an identity
c)
it is associative, has an inverse
d)
it is commutative, has an identity, has an inverse
92.
A group (G, *) is said to be Abelian if * is ___. 
a)
associative
b)
commutative
c)
distributive
d)
commutative and distributive
93.
Which of the followings is NOT an Abelian group under addition?  
a)
Real numbers, R
b)
Positive integer, Z+
c)
rational numbers, Q
d)
Integers, Z
94.
Monoid has the following properties:  
a)
it is commutative, has an identity element 
b)
it is associative, has an identity element 
c)
it has identity element
d)
it has an inverse
95.

Which property states that the operation in a group must produce an element that is also in the group?

a)

Associativity

b)

Closure

c)

Inverse Element

d)

Identity Element

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