MAT 1350 Mathematics for Liberal Arts Final Exam | Complete Practice Questions & Detailed Solutions

MAT 1350 Mathematics for Liberal Arts Final Exam | Complete
Practice Questions & Detailed Solutions


Question 1
If set 𝐴 = {2,4,6,8,10}and set 𝐵 = {4,8,12}, what is the cardinality of
the union of sets 𝐴and 𝐵, denoted as 𝑛(𝐴 ∪ 𝐵)?
• A. 5
• B. 6
• C. 7
• D. 8
• Correct Answer: B
• Detailed Rationale: The union 𝐴 ∪ 𝐵combines all unique
elements from both sets: {2,4,6,8,10,12}. Counting these
elements gives a cardinality of 6.
Question 2
Given universal set 𝑈 = {1,2,3,4,5,6,7,8,9,10}and set 𝐴 = {1,3,5,7,9},
what is the complement of set 𝐴, written as 𝐴′?
• A. {1,3,5,7,9}
• B. {2,4,6,8,10}
• C. {2,3,5,7}
• D. ∅
• Correct Answer: B

, • Detailed Rationale: The complement 𝐴′contains all elements in
the universal set 𝑈that are not in set 𝐴. Removing the odd
numbers from 1to 10leaves the even numbers: {2,4,6,8,10}.
Question 3
In a survey of 100college students, 60take Psychology, 45take Statistics,
and 20take both. How many students take neither Psychology nor
Statistics?
• A. 10
• B. 15
• C. 25
• D. 35
• Correct Answer: B
• Detailed Rationale: Using the Principle of Inclusion-Exclusion, the
number of students taking at least one subject is 𝑛(𝑃 ∪ 𝑆) =
𝑛(𝑃) + 𝑛(𝑆) − 𝑛(𝑃 ∩ 𝑆) = 60 + 45 − 20 = 85. The number
taking neither is the total minus this union: 100 − 85 = 15.
Question 4
Which of the following sets represents the intersection of 𝐴 =
{𝑥 ∣ 𝑥 is an even integer between 1 and 10}and 𝐵 = {𝑥 ∣
𝑥 is a multiple of 3 up to 10}?
• A. {6}
• B. {2,4,6,8}
• C. {3,6,9}

, • D. {2,3,4,6,8,9}
• Correct Answer: A
• Detailed Rationale: Set 𝐴 = {2,4,6,8}and Set 𝐵 = {3,6,9}. The
intersection 𝐴 ∩ 𝐵contains elements common to both sets, which
is only {6}.
Question 5
According to De Morgan's Laws for sets, which of the following is
equivalent to (𝐴 ∪ 𝐵)′?
• A. 𝐴′ ∪ 𝐵′
• B. 𝐴′ ∩ 𝐵′
• C. (𝐴 ∩ 𝐵)′
• D. 𝐴 ∪ 𝐵′
• Correct Answer: 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 6
Let 𝑝represent "It is raining" and 𝑞represent "The grass is wet." What is
the symbolic translation of "It is not raining or the grass is wet"?
• A. ∼ 𝑝 ∧ 𝑞
• B. ∼ 𝑝 ∨ 𝑞
• C. ∼ (𝑝 ∨ 𝑞 )
• D. 𝑝 ∨∼ 𝑞

, • Correct Answer: B
• Detailed Rationale: "It is not raining" translates to ∼ 𝑝. "Or"
translates to the disjunction symbol ∨. "The grass is wet"
translates to 𝑞. Thus, the expression is ∼ 𝑝 ∨ 𝑞.
Question 7
What is the contrapositive of the conditional statement "If you pass the
final exam, then you pass the course"?
• A. If you do not pass the final exam, then you do not pass the
course.
• B. If you pass the course, then you pass the final exam.
• C. If you do not pass the course, then you do not pass the final
exam.
• D. If you do not pass the course, then you pass the final exam.
• Correct Answer: C
• Detailed Rationale: The contrapositive of 𝑝 → 𝑞is ∼ 𝑞 →∼ 𝑝.
Negating and swapping both clauses yields: "If you do not pass the
course, then you do not pass the final exam."
Question 8
Which of the following truth table outcomes describes a tautology?
• A. All false values in the final column
• B. All true values in the final column
• C. A mix of true and false values in the final column
• D. Only alternating true and false values