BTCS-220 Discrete Mathematics Final Exam | Complete
Practice QUESTIONs, Verified Answers & Detailed Solutions
(2026/2027)
QUESTION 1
Which of the following propositional equivalences correctly describes
the implication 𝑝 → 𝑞?
A. ¬𝑝 ∨ 𝑞
B. 𝑝 ∨ ¬𝑞
C. ¬𝑝 ∧ 𝑞
D. ¬𝑞 → ¬𝑝
Correct Answer: A. ¬𝑝 ∨ 𝑞
Detailed Rationale: The conditional statement 𝑝 → 𝑞 is logically
equivalent to "not 𝑝 or 𝑞" (¬𝑝 ∨ 𝑞). Note that option D is its
contrapositive (¬𝑞 → ¬𝑝), which is also logically equivalent, but option
A is the standard disjunctive definition of implication.
QUESTION 2
What is the negation of the quantified statement ∀𝑥 𝑃(𝑥 )?
A. ∀𝑥 ¬𝑃(𝑥 )
B. ∃𝑥 ¬𝑃(𝑥 )
C. ∃𝑥 𝑃(𝑥 )
D. ¬∀𝑥 ¬𝑃(𝑥 )
,Correct Answer: B. ∃𝑥 ¬𝑃(𝑥 )
Detailed Rationale: According to the rules of quantifiers, negating a
universal statement changes the quantifier to existential and negates
the predicate: ¬(∀𝑥 𝑃(𝑥 )) ≡ ∃𝑥 ¬𝑃(𝑥 ).
QUESTION 3
If a set 𝐴 has a cardinality of 4 (|𝐴| = 4), what is the cardinality of its
power set 𝒫 (𝐴)?
A. 4
B. 8
C. 16
D. 32
Correct Answer: C. 16
Detailed Rationale: The power set of a finite set with 𝑛 elements
contains 2𝑛 subsets. For |𝐴| = 4, the cardinality of the power set is
24 = 16.
QUESTION 4
According to De Morgan's Laws for sets, what is the complement of the
union of two sets, (𝐴 ∪ 𝐵)′?
A. 𝐴′ ∪ 𝐵′
B. 𝐴′ ∩ 𝐵′
C. (𝐴 ∩ 𝐵)′
D. 𝐴 ∪ 𝐵′
Correct Answer: B. 𝐴′ ∩ 𝐵′
,Detailed Rationale: De Morgan's laws state that the complement of a
union is the intersection of the complements: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′.
QUESTION 5
A relation 𝑅 on a set 𝐴 is called an equivalence relation if it satisfies
which three properties?
A. Reflexive, Symmetric, Transitive
B. Reflexive, Antisymmetric, Transitive
C. Irreflexive, Symmetric, Transitive
D. Reflexive, Symmetric, Asymmetric
Correct Answer: A. Reflexive, Symmetric, Transitive
Detailed Rationale: By definition, an equivalence relation must be
reflexive (𝑎𝑅𝑎), symmetric (𝑎𝑅𝑏 ⟹ 𝑏𝑅𝑎), and transitive (𝑎𝑅𝑏 and
𝑏𝑅𝑐 ⟹ 𝑎𝑅𝑐).
QUESTION 6
A function 𝑓: 𝐴 → 𝐵 is called bijective if it is:
A. Only injective (one-to-one)
B. Only surjective (onto)
C. Both injective and surjective
D. Neither injective nor surjective
Correct Answer: C. Both injective and surjective
Detailed Rationale: A bijection is a one-to-one and onto function,
meaning every element in the codomain is mapped to by exactly one
element in the domain.
, QUESTION 7
In how many ways can 5 distinct books be arranged on a shelf?
A. 24
B. 60
C. 120
D. 720
Correct Answer: C. 120
Detailed Rationale: The number of ways to arrange 𝑛 distinct objects in
a linear order is 𝑛! (factorial). For 𝑛 = 5, 5! = 5 × 4 × 3 × 2 × 1 =
120.
QUESTION 8
What is the value of the binomial coefficient (10
3
)?
A. 720
B. 120
C. 210
D. 360
Correct Answer: B. 120
𝑛!
Detailed Rationale: The formula for combinations is (𝑛𝑟) = .
𝑟!(𝑛−𝑟)!
10×9×8 720
Evaluating for (10
3
)= = = 120.
3×2×1 6
QUESTION 9
Practice QUESTIONs, Verified Answers & Detailed Solutions
(2026/2027)
QUESTION 1
Which of the following propositional equivalences correctly describes
the implication 𝑝 → 𝑞?
A. ¬𝑝 ∨ 𝑞
B. 𝑝 ∨ ¬𝑞
C. ¬𝑝 ∧ 𝑞
D. ¬𝑞 → ¬𝑝
Correct Answer: A. ¬𝑝 ∨ 𝑞
Detailed Rationale: The conditional statement 𝑝 → 𝑞 is logically
equivalent to "not 𝑝 or 𝑞" (¬𝑝 ∨ 𝑞). Note that option D is its
contrapositive (¬𝑞 → ¬𝑝), which is also logically equivalent, but option
A is the standard disjunctive definition of implication.
QUESTION 2
What is the negation of the quantified statement ∀𝑥 𝑃(𝑥 )?
A. ∀𝑥 ¬𝑃(𝑥 )
B. ∃𝑥 ¬𝑃(𝑥 )
C. ∃𝑥 𝑃(𝑥 )
D. ¬∀𝑥 ¬𝑃(𝑥 )
,Correct Answer: B. ∃𝑥 ¬𝑃(𝑥 )
Detailed Rationale: According to the rules of quantifiers, negating a
universal statement changes the quantifier to existential and negates
the predicate: ¬(∀𝑥 𝑃(𝑥 )) ≡ ∃𝑥 ¬𝑃(𝑥 ).
QUESTION 3
If a set 𝐴 has a cardinality of 4 (|𝐴| = 4), what is the cardinality of its
power set 𝒫 (𝐴)?
A. 4
B. 8
C. 16
D. 32
Correct Answer: C. 16
Detailed Rationale: The power set of a finite set with 𝑛 elements
contains 2𝑛 subsets. For |𝐴| = 4, the cardinality of the power set is
24 = 16.
QUESTION 4
According to De Morgan's Laws for sets, what is the complement of the
union of two sets, (𝐴 ∪ 𝐵)′?
A. 𝐴′ ∪ 𝐵′
B. 𝐴′ ∩ 𝐵′
C. (𝐴 ∩ 𝐵)′
D. 𝐴 ∪ 𝐵′
Correct Answer: B. 𝐴′ ∩ 𝐵′
,Detailed Rationale: De Morgan's laws state that the complement of a
union is the intersection of the complements: (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′.
QUESTION 5
A relation 𝑅 on a set 𝐴 is called an equivalence relation if it satisfies
which three properties?
A. Reflexive, Symmetric, Transitive
B. Reflexive, Antisymmetric, Transitive
C. Irreflexive, Symmetric, Transitive
D. Reflexive, Symmetric, Asymmetric
Correct Answer: A. Reflexive, Symmetric, Transitive
Detailed Rationale: By definition, an equivalence relation must be
reflexive (𝑎𝑅𝑎), symmetric (𝑎𝑅𝑏 ⟹ 𝑏𝑅𝑎), and transitive (𝑎𝑅𝑏 and
𝑏𝑅𝑐 ⟹ 𝑎𝑅𝑐).
QUESTION 6
A function 𝑓: 𝐴 → 𝐵 is called bijective if it is:
A. Only injective (one-to-one)
B. Only surjective (onto)
C. Both injective and surjective
D. Neither injective nor surjective
Correct Answer: C. Both injective and surjective
Detailed Rationale: A bijection is a one-to-one and onto function,
meaning every element in the codomain is mapped to by exactly one
element in the domain.
, QUESTION 7
In how many ways can 5 distinct books be arranged on a shelf?
A. 24
B. 60
C. 120
D. 720
Correct Answer: C. 120
Detailed Rationale: The number of ways to arrange 𝑛 distinct objects in
a linear order is 𝑛! (factorial). For 𝑛 = 5, 5! = 5 × 4 × 3 × 2 × 1 =
120.
QUESTION 8
What is the value of the binomial coefficient (10
3
)?
A. 720
B. 120
C. 210
D. 360
Correct Answer: B. 120
𝑛!
Detailed Rationale: The formula for combinations is (𝑛𝑟) = .
𝑟!(𝑛−𝑟)!
10×9×8 720
Evaluating for (10
3
)= = = 120.
3×2×1 6
QUESTION 9
