ECAT Mathematics
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ECAT Mathematics
Chapter 2: Sets, Functions and Groups
90 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
A set is a well-defined collection of:
A
Similar items
B
Distinct objects
C
Only numbers
D
Only letters
1
The empty set has:
A
One element
B
No elements
C
All elements
D
Infinite elements
2
Number of subsets of {a,b,c}:
A
3
B
6
C
8
D
9
3
A∪B is:
A
Intersection
B
Union
C
Complement
D
Difference
4
A∩B is:
A
Union
B
Intersection
C
Complement
D
Difference
5
If A={1,2,3}, B={2,3,4}: A∩B=
A
{{1,2,3,4}}
B
{2,3}
C
{1,4}
D
{1}
6
If A={1,2,3}, B={2,3,4}: A∪B=
A
{{1,2,3,4}}
B
{2,3}
C
{1,4}
D
{1,2,3}
7
A' (complement) contains:
A
All elements in A
B
All elements in U not in A
C
A∩B
D
A∪B
8
(A∪B)' =
A
A'∩B'
B
A'∪B'
C
A∩B
D
A∪B
9
(A∩B)' =
A
A'∩B'
B
A'∪B'
C
A∩B
D
A∪B
10
Injective (one-to-one) function: f(x₁)=f(x₂) implies:
A
x₁=x₂
B
f is onto
C
f(x)=x
D
f is symmetric
11
Surjective (onto): every element of B has:
A
No pre-image
B
At least one pre-image in A
C
Exactly one pre-image
D
None
12
Bijective =
A
Only injective
B
Only surjective
C
Injective and surjective
D
Neither
13
Domain of f(x)=1/(x−2):
A
All reals
B
All reals except 2
C
x>2
D
x<2
14
Range of f(x)=x² (x∈R):
A
All reals
B
[0,∞)
C
(0,∞)
D
(−∞,0]
15
f(x)=2x+1, g(x)=x−3. f(g(x))=
A
2x−5
B
2x+1
C
2x−4
D
2x−3
16
Identity function:
A
f(x)=0
B
f(x)=1
C
f(x)=x
D
f(x)=−x
17
Group (G,*) requires:
A
Closure, associativity, identity, inverse
B
Commutativity only
C
Closure only
D
Closure and commutativity
18
Abelian group is also:
A
Associative
B
Commutative
C
Closed
D
Cyclic
19
(Z,+) is a:
A
Field
B
Ring but not group
C
Group
D
Semigroup only
20
n(A∪B)=n(A)+n(B)−n(A∩B) is:
A
Inclusion-exclusion principle
B
De Morgan law
C
Complement law
D
Absorption law
21
n(A)=5, n(B)=3, n(A∩B)=2: n(A∪B)=
A
8
B
6
C
10
D
4
22
Power set of {a,b}:
A
{{a,b}}
B
{{a},{b}}
C
{{},{a},{b},{a,b}}
D
{{a,b},{}}
23
A⊆B and B⊆A implies:
A
A=B
B
A≠B
C
A={}
D
B={}
24
A relation is a function if:
A
Each domain element has exactly one image
B
Range=domain
C
Relation is symmetric
D
Relation is reflexive
25
f⁻¹ exists if f is:
A
Only injective
B
Only surjective
C
Bijective
D
Constant
26
f(x)=3x−2: f⁻¹(x)=
A
(x+2)/3
B
(x−2)/3
C
3x+2
D
(x−3)/2
27
Order of group G is:
A
Number of elements
B
Identity element
C
Number of subgroups
D
Largest element
28
{1,−1} under multiplication:
A
Not a group
B
Group of order 2
C
Infinite group
D
A ring
29
Subgroup H of G satisfies:
A
Non-empty, closed, contains inverses
B
H=G
C
H has one element
D
H is commutative
30
A×B contains:
A
Common elements
B
All ordered pairs (a,b): a∈A, b∈B
C
A∪B
D
A−B
31
|A×B| when |A|=m, |B|=n:
A
m+n
B
mn
C
m−n
D
mⁿ
32
Symmetric difference A△B:
A
A∩B
B
A∪B
C
(A−B)∪(B−A)
D
A∩Bᶜ
33
Even function: f(−x)=
A
f(x)
B
−f(x)
C
0
D
1
34
Odd function: f(−x)=
A
f(x)
B
−f(x)
C
f(x)²
D
0
35
(f∘g)(x)=
A
f(x)g(x)
B
f(g(x))
C
g(f(x))
D
f(x)+g(x)
36
Constant function f(x)=c is:
A
One-to-one
B
Onto
C
Neither injective nor surjective (generally)
D
Bijective
37
Identity for (Z,+) is:
A
0
B
1
C
−1
D
None
38
Identity element of a group is:
A
Unique
B
May be multiple
C
Not needed
D
Always the largest
39
Each group element's inverse is:
A
Unique
B
Not unique
C
The element itself always
D
Always 0
40
A={1,2,3,4,5}, B={2,4}. A−B=
A
{{1,3,5}}
B
{2,4}
C
{1,2,3,4,5}
D
{1,3,4,5}
41
f: A→B means:
A
f maps B to A
B
f maps A to B
C
A is function
D
B equals A
42
Reflexive relation: (a,a)∈R for all a∈A means:
A
Symmetric
B
Transitive
C
Reflexive
D
Equivalence
43
Transitive: if (a,b)∈R and (b,c)∈R then:
A
(a,a)∈R
B
(b,a)∈R
C
(a,c)∈R
D
(c,a)∈R
44
Equivalence relation is:
A
Reflexive, symmetric, transitive
B
Only reflexive
C
Only symmetric
D
Only transitive
45
Lagrange's theorem: order of subgroup divides:
A
Identity order
B
Order of G
C
Number of subgroups
D
Element order
46
Zₙ under addition mod n is:
A
Cyclic and abelian
B
Never abelian
C
Non-commutative
D
Infinite
47
g(f(x)) where f(x)=x²−1, g(x)=x+1:
A
x²
B
x²+1
C
x³−x
D
x²−1
48
A∩(B∪C)=
A
(A∩B)∪(A∩C)
B
(A∪B)∩(A∪C)
C
A∩B∩C
D
A∪B∪C
49
f:R→R by f(x)=eˣ is:
A
Onto
B
One-to-one but not onto
C
Bijective
D
Neither
50
Functions from {a,b} to {1,2,3}:
A
6
B
8
C
9
D
5
51
Proper subsets of 4-element set:
A
16
B
15
C
14
D
8
52
Domain of f(x)=√(x−3):
A
x>3
B
x≥3
C
x<3
D
All reals
53
(f⁻¹)⁻¹=
A
f⁻¹
B
f
C
Identity
D
Constant
54
{0} under multiplication is:
A
A group
B
Not a group (no inverse for 0)
C
Infinite set
D
A field
55
(ab)⁻¹=
A
a⁻¹b⁻¹
B
b⁻¹a⁻¹
C
ab
D
ba
56
2×2 invertible matrices under multiplication:
A
Commutative group
B
Non-commutative group
C
Not a group
D
Ring only
57
Disjoint sets A∩B=
A
A
B
B
C
{}
D
A∪B
58
Partition of A: subsets are:
A
Overlapping and covering A
B
Disjoint, non-empty, covering A
C
All empty
D
All equal
59
Absorption law: A∪(A∩B)=
A
B
B
A
C
A∩B
D
A∪B
60
If f and g are bijections, f∘g is:
A
Not a function
B
A bijection
C
Only injective
D
Only surjective
61
ab=ac in group G implies:
A
b≠c
B
b=c
C
a=b=c
D
a=0
62
Group with one element:
A
Trivial group
B
Cyclic group
C
Abelian group
D
Infinite group
63
Idempotent element satisfies:
A
a*a=e
B
a*a=a
C
a*b=b*a
D
a*b=e
64
Cyclic subgroup generated by a is:
A
Normal subgroup
B
Intersection of all subgroups containing a
C
Center of G
D
Coset
65
n(A')=
A
n(A)
B
n(U)−n(A)
C
n(U)+n(A)
D
n(A)/n(U)
66
f(x)=|x| is:
A
Bijective
B
One-to-one
C
Onto
D
Neither injective nor surjective
67
Equal sets have:
A
Same cardinality only
B
Exactly same elements
C
Same type
D
Elements from same universe
68
Universal set U:
A
Always infinite
B
Contains all sets under consideration
C
Is always proper subset
D
Is empty set
69
{x: x even natural number <10}=
A
{{2,4,6,8,10}}
B
{2,4,6,8}
C
{0,2,4,6,8}
D
{1,3,5,7,9}
70
Cyclic group is always:
A
Finite
B
Infinite
C
Abelian
D
Non-abelian
71
Max |A∩B| when |A|=4, |B|=6:
A
4
B
6
C
10
D
2
72
f∘f=identity means f is:
A
Identity
B
Involution (self-inverse)
C
Constant
D
Bijection
73
All bijections A→A form:
A
Power set
B
Symmetric group S_A
C
Cyclic group
D
Normal group
74
Disjoint A,B: n(A∪B)=
A
n(A)−n(B)
B
n(A)×n(B)
C
n(A)+n(B)
D
n(A)/n(B)
75
Which is NOT a function?
A
{{(1,2),(2,3)}}
B
{(1,2),(1,3)}
C
{(1,1),(2,2)}
D
{(3,5)}
76
aⁿ=e for some n>0 means a has:
A
Infinite order
B
Finite order n
C
Order 0
D
No order
77
f(x)=2x+3: f⁻¹(x)=
A
(x−3)/2
B
2x−3
C
(x+3)/2
D
(x+2)/3
78
Venn diagrams represent:
A
Equations
B
Sets and their relations
C
Trig functions
D
Algebra
79
|A×A| where |A|=3:
A
3
B
6
C
9
D
8
80
Identity of (Q,+):
A
1
B
0
C
−1
D
undefined
81
Identity of (Q−{0},×):
A
0
B
−1
C
1
D
1/2
82
Additive inverse of −5 in (Z,+):
A
5
B
−5
C
1/5
D
0
83
Multiplicative inverse of 3 in (Q−{0},×):
A
3
B
−3
C
1/3
D
−1/3
84
Inverse of f(x)=x³:
A
x^(1/3)
B
x³
C
−x³
D
3x
85
Monoid is a semigroup with:
A
Inverses
B
An identity element
C
Commutativity
D
Distributivity
86
(Z,−) subtraction is NOT a group because:
A
Not closed
B
Not associative
C
No identity
D
No inverses
87
Proper subset of A:
A
Equals A
B
Strictly contained in A
C
Contains extra elements
D
Is empty
88
n(A×B) where A={a,b,c}, B={1,2}:
A
5
B
6
C
8
D
4
89
Range of f(x)=sin(x):
A
All reals
B
[−1,1]
C
[0,1]
D
(0,∞)
0
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