MAT3702 ABSTRACT ALGEBRA — REVISION
PACK EXAM 4 2026 PAST EXAM
QUESTIONS & ANSWERS WITH WORKED
EXPLANATIONS
SECTION A: SETS, FUNCTIONS & RELATIONS
Question 1: Set Operations
Let A={1,2,3,4}A={1,2,3,4} and B={3,4,5,6}B={3,4,5,6}. Determine
A∪BA∪B.
A. {3,4}{3,4}
B. {1,2,5,6}{1,2,5,6}
C. {1,2,3,4,5,6}{1,2,3,4,5,6}
D. ∅∅
Answer: C. {1,2,3,4,5,6}{1,2,3,4,5,6}.
Solution: The union contains every element belonging to either
set, without repetition.
Question 2: Set Intersection
Using the sets in Question 1, determine A∩BA∩B.
A. {1,2}{1,2}
B. {3,4}{3,4}
,C. {5,6}{5,6}
D. {1,2,3,4,5,6}{1,2,3,4,5,6}
Answer: B. {3,4}{3,4}.
Solution: The intersection contains only elements common to
both sets.
Question 3: Injective Functions
A function f:A→Bf:A→B is injective if:
A. Every element of BB has a preimage.
B. Every element of AA maps to itself.
C. Different elements of AA always have different images.
D. The domain and codomain are identical.
Answer: C. Different elements of AA always have different
images.
Solution: Injectivity means f(x)=f(y)⇒x=yf(x)=f(y)⇒x=y.
Equivalently, distinct domain elements cannot map to the same
codomain element.
Question 4: Surjective Functions
A function f:A→Bf:A→B is surjective when:
A. Every element of BB is the image of at least one element of AA.
B. Every element of AA has two images.
, C. ff is necessarily constant.
D. The domain is empty.
Answer: A. Every element of BB is the image of at least one
element of AA.
Solution: Surjectivity means that for every b∈Bb∈B, there exists
a∈Aa∈A such that f(a)=bf(a)=b.
Question 5: Equivalence Relation
Define a relation on ZZ by a∼ba∼b if and only if 5∣(a−b)5∣(a−b).
Show that this is an equivalence relation.
Solution:
Reflexive: a−a=0a−a=0 is divisible by 55.
Symmetric: If 5∣(a−b)5∣(a−b), then 5∣(b−a)5∣(b−a).
Transitive: If 5∣(a−b)5∣(a−b) and 5∣(b−c)5∣(b−c), then 5∣(a−c)5∣(a−c).
Thus ∼∼ is an equivalence relation.
Question 6: De Morgan's Laws
Which identity is correct for sets AA and BB in a universal set UU?
A. (A∪B)c=Ac∩Bc(A∪B)c=Ac∩Bc
B. (A∪B)c=Ac∪Bc(A∪B)c=Ac∪Bc
PACK EXAM 4 2026 PAST EXAM
QUESTIONS & ANSWERS WITH WORKED
EXPLANATIONS
SECTION A: SETS, FUNCTIONS & RELATIONS
Question 1: Set Operations
Let A={1,2,3,4}A={1,2,3,4} and B={3,4,5,6}B={3,4,5,6}. Determine
A∪BA∪B.
A. {3,4}{3,4}
B. {1,2,5,6}{1,2,5,6}
C. {1,2,3,4,5,6}{1,2,3,4,5,6}
D. ∅∅
Answer: C. {1,2,3,4,5,6}{1,2,3,4,5,6}.
Solution: The union contains every element belonging to either
set, without repetition.
Question 2: Set Intersection
Using the sets in Question 1, determine A∩BA∩B.
A. {1,2}{1,2}
B. {3,4}{3,4}
,C. {5,6}{5,6}
D. {1,2,3,4,5,6}{1,2,3,4,5,6}
Answer: B. {3,4}{3,4}.
Solution: The intersection contains only elements common to
both sets.
Question 3: Injective Functions
A function f:A→Bf:A→B is injective if:
A. Every element of BB has a preimage.
B. Every element of AA maps to itself.
C. Different elements of AA always have different images.
D. The domain and codomain are identical.
Answer: C. Different elements of AA always have different
images.
Solution: Injectivity means f(x)=f(y)⇒x=yf(x)=f(y)⇒x=y.
Equivalently, distinct domain elements cannot map to the same
codomain element.
Question 4: Surjective Functions
A function f:A→Bf:A→B is surjective when:
A. Every element of BB is the image of at least one element of AA.
B. Every element of AA has two images.
, C. ff is necessarily constant.
D. The domain is empty.
Answer: A. Every element of BB is the image of at least one
element of AA.
Solution: Surjectivity means that for every b∈Bb∈B, there exists
a∈Aa∈A such that f(a)=bf(a)=b.
Question 5: Equivalence Relation
Define a relation on ZZ by a∼ba∼b if and only if 5∣(a−b)5∣(a−b).
Show that this is an equivalence relation.
Solution:
Reflexive: a−a=0a−a=0 is divisible by 55.
Symmetric: If 5∣(a−b)5∣(a−b), then 5∣(b−a)5∣(b−a).
Transitive: If 5∣(a−b)5∣(a−b) and 5∣(b−c)5∣(b−c), then 5∣(a−c)5∣(a−c).
Thus ∼∼ is an equivalence relation.
Question 6: De Morgan's Laws
Which identity is correct for sets AA and BB in a universal set UU?
A. (A∪B)c=Ac∩Bc(A∪B)c=Ac∩Bc
B. (A∪B)c=Ac∪Bc(A∪B)c=Ac∪Bc
