Abstract Algebra MAT3702 – Course Notes and Exam Preparation

MAT3702 Abstract Algebra — Revision
Pack (Exam 3)


Section A: Sets, Functions and Relations (Questions 1–35)

1. 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. ∅∅

Correct Answer: C
Rationale: The union contains every element belonging to either set,
without repetition.

2. 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}

Correct Answer: B
Rationale: The intersection contains only elements common to both sets.

3. 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.

Correct Answer: C
Rationale: Injectivity means f(x)=f(y)⇒x=yf(x)=f(y)⇒x=y; distinct domain
elements cannot map to the same codomain element.

4. 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.

Correct Answer: A
Rationale: Surjectivity means that for every b∈Bb∈B, there exists a∈Aa∈A
such that f(a)=bf(a)=b.

5. Which identity is De Morgan’s law 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
C. (A∩B)c=Ac∩Bc(A∩B)c=Ac∩Bc
D. A∪Ac=AA∪Ac=A

Correct Answer: A
Rationale: The complement of a union is the intersection of the
complements. The other law is (A∩B)c=Ac∪Bc(A∩B)c=Ac∪Bc.

6. How many elements are in the power set of a finite set containing
four elements?
A. 4
B. 8

,C. 16
D. 24

Correct Answer: C
Rationale: A set with nn elements has 2n2n subsets. Therefore, the power
set has 24=1624=16 elements.

7. Which pair of integers is relatively prime?
A. 8 and 12
B. 9 and 15
C. 14 and 21
D. 8 and 15

Correct Answer: D
Rationale: The greatest common divisor of 8 and 15 is 1, so the numbers
are relatively prime.

8. The relation RR on ZZ defined by aRbaRb iff 5∣(a−b)5∣(a−b) is:
A. Reflexive only
B. Symmetric only
C. An equivalence relation
D. Not transitive

Correct Answer: C
Rationale: Since a−a=0a−a=0 is divisible by 5, the relation is reflexive. If
5∣(a−b)5∣(a−b), then 5∣(b−a)5∣(b−a), so it is symmetric. If 5∣(a−b)5∣(a−b) and
5∣(b−c)5∣(b−c), then 5∣(a−c)5∣(a−c), so it is transitive.

9. Let A={1,2,3}A={1,2,3}. How many subsets does AA have?
A. 3
B. 6
C. 8
D. 9

, Correct Answer: C
Rationale: A set with nn elements has 2n2n subsets. For n=3n=3, 23=823=8.

10. If f:A→Bf:A→B and g:B→Cg:B→C are both bijections, then g∘fg∘f is:
A. Injective only
B. Surjective only
C. A bijection
D. Not necessarily a function

Correct Answer: C
Rationale: The composition of two bijections is a bijection.

11. The Cartesian product A×BA×B of sets AA and BB is:
A. The set of all subsets of A∪BA∪B
B. The set of all ordered pairs (a,b)(a,b) with a∈Aa∈A and b∈Bb∈B
C. The set of common elements of AA and BB
D. The union of AA and BB

Correct Answer: B
Rationale: A×B={(a,b)∣a∈A, b∈B}A×B={(a,b)∣a∈A, b∈B}.

12. A relation RR on a set SS is antisymmetric if:
A. aRbaRb implies bRabRa
B. aRbaRb and bRabRa imply a=ba=b
C. aRaaRa for all aa
D. aRbaRb and bRcbRc imply aRcaRc

Correct Answer: B
Rationale: Antisymmetry prohibits distinct elements from being related in
both directions.

13. Which of the following is an equivalence relation?
A. “Less than” on RR
B. “Divides” on Z+Z+