ma3001_2026T2_Q2_NA.pdf
Discrete Mathematics · Quiz 2 · May 2026
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Question 2 MSQ · 4.0 marks
Consider the following truth table:
[[IMAGE:410d6c203b078554_2_2]]
[[IMAGE:410d6c203b078554_2_3]]
[[IMAGE:410d6c203b078554_2_4]]
[[IMAGE:410d6c203b078554_2_5]]
[[IMAGE:410d6c203b078554_2_6]]
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Question 3 MCQ · 4.0 marks
Use axioms of inference to determine whether the conclusion inferred from the hypothesis is
logically valid or not.
The hypotheses are:
1. It is not raining or Yvette has her umbrella.
2. Yvette does not have her umbrella or she does not get wet.
3. It is raining or Yvette does not get wet.
Thus, the conclusion is: Yvette does not get wet.
The conclusion is valid.
The conclusion is invalid.
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Question 4 MCQ · 4.0 marks
[[IMAGE:410d6c203b078554_3_7]]
True
False
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Question 5 MCQ · 2.0 marks
[[IMAGE:410d6c203b078554_4_8]]
[[IMAGE:410d6c203b078554_4_9]]
[[IMAGE:410d6c203b078554_4_10]]
[[IMAGE:410d6c203b078554_4_11]]
[[IMAGE:410d6c203b078554_4_12]]
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Question 6 MCQ · 2.0 marks
Consider the following proof to show that every positive integer can be written as a sum of a
subset of integers that are powers of [[IMAGE:410d6c203b078554_4_13]] . For example, the integers [[IMAGE:410d6c203b078554_4_14]] and [[IMAGE:410d6c203b078554_4_15]] are powers
of [[IMAGE:410d6c203b078554_4_16]] , and the integer [[IMAGE:410d6c203b078554_4_17]] can be written as a sum of two integers that are
powers of [[IMAGE:410d6c203b078554_4_18]] .
Proof. Let [[IMAGE:410d6c203b078554_4_19]] be the statement that the positive integer [[IMAGE:410d6c203b078554_4_20]] can be written as a sum of a subset of
distinct powers of [[IMAGE:410d6c203b078554_4_21]] .
Base Case: When [[IMAGE:410d6c203b078554_4_22]] , then [[IMAGE:410d6c203b078554_4_23]] . Hence [[IMAGE:410d6c203b078554_4_24]] is true.
Induction Hypothesis: ____________. We have to show that [[IMAGE:410d6c203b078554_4_25]] is true.
Inductive Proof:
Case 1: If [[IMAGE:410d6c203b078554_4_26]] is a power of [[IMAGE:410d6c203b078554_4_27]] , then let [[IMAGE:410d6c203b078554_4_28]] and hence, [[IMAGE:410d6c203b078554_4_29]] is true.
Case 2: If [[IMAGE:410d6c203b078554_4_30]] is not a power of [[IMAGE:410d6c203b078554_4_31]] . Let [[IMAGE:410d6c203b078554_4_32]] be the largest power of [[IMAGE:410d6c203b078554_4_33]] less than [[IMAGE:410d6c203b078554_4_34]] , i.e.,
[[IMAGE:410d6c203b078554_4_35]] . Let [[IMAGE:410d6c203b078554_4_36]] . Then, [[IMAGE:410d6c203b078554_4_37]] is the integer such that
[[IMAGE:410d6c203b078554_4_38]] . Hence, by the induction hypothesis, [[IMAGE:410d6c203b078554_4_39]] is true, i.e., [[IMAGE:410d6c203b078554_4_40]] can be
written as a sum of a subset of distinct powers of [[IMAGE:410d6c203b078554_4_41]] . Therefore, [[IMAGE:410d6c203b078554_4_42]] , where both
terms are sums of distinct powers of [[IMAGE:410d6c203b078554_5_43]] . Hence [[IMAGE:410d6c203b078554_5_44]] is true.
Which of the following is the correct induction hypothesis for this proof?
Assume [[IMAGE:410d6c203b078554_5_45]] be true, where [[IMAGE:410d6c203b078554_5_46]]
Assume [[IMAGE:410d6c203b078554_5_47]] is true for every positive integer [[IMAGE:410d6c203b078554_5_48]] , where [[IMAGE:410d6c203b078554_5_49]]
Assume [[IMAGE:410d6c203b078554_5_50]] be true, where [[IMAGE:410d6c203b078554_5_51]]
Assume [[IMAGE:410d6c203b078554_5_52]] is true for every positive integer [[IMAGE:410d6c203b078554_5_53]] , where [[IMAGE:410d6c203b078554_5_54]]
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Question 7 MCQ · 2.0 marks
Which of the following is a solution to the recurrence relation [[IMAGE:410d6c203b078554_5_55]] , where [[IMAGE:410d6c203b078554_5_56]] ?
[[IMAGE:410d6c203b078554_5_57]]
[[IMAGE:410d6c203b078554_5_58]]
[[IMAGE:410d6c203b078554_5_59]]
None of these
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Question 8 NAT · 3.0 marks
Calculate the value of
[[IMAGE:410d6c203b078554_6_60]]
.
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Question 9 NAT · 3.0 marks
Among a group of [[IMAGE:410d6c203b078554_6_61]] (not necessarily consecutive) integers, at least how many integers must have
the same remainder when divided by [[IMAGE:410d6c203b078554_6_62]] ?
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Question 10 MSQ · 3.0 marks
[[IMAGE:410d6c203b078554_6_63]]
[[IMAGE:410d6c203b078554_6_64]]
[[IMAGE:410d6c203b078554_6_65]]
[[IMAGE:410d6c203b078554_7_66]]
[[IMAGE:410d6c203b078554_7_67]]
[[IMAGE:410d6c203b078554_7_68]]
A published solution is not available for this question yet.
Question 11 MSQ · 3.0 marks
How many ways are there to travel in the [[IMAGE:410d6c203b078554_7_69]] -dimensional space from the origin [[IMAGE:410d6c203b078554_7_70]] to the
point [[IMAGE:410d6c203b078554_7_71]] by taking one unit steps in the positive [[IMAGE:410d6c203b078554_7_72]] , positive [[IMAGE:410d6c203b078554_7_73]] , or positive [[IMAGE:410d6c203b078554_7_74]] direction?
[[IMAGE:410d6c203b078554_7_75]]
[[IMAGE:410d6c203b078554_7_76]]
[[IMAGE:410d6c203b078554_7_77]]
[[IMAGE:410d6c203b078554_7_78]]
A published solution is not available for this question yet.
Question 12 MCQ · 3.0 marks
Consider the following countable set:
[[IMAGE:410d6c203b078554_7_79]]
Which of the following statements establishes a one-to-one correspondence between the set of
natural numbers [[IMAGE:410d6c203b078554_7_80]] and [[IMAGE:410d6c203b078554_7_81]] ?
For each
[[IMAGE:410d6c203b078554_8_82]] , assign the bit string containing [[IMAGE:410d6c203b078554_8_83]] 1s.
For each [[IMAGE:410d6c203b078554_8_84]] , assign the natural number whose binary representation is [[IMAGE:410d6c203b078554_8_85]] .
[[IMAGE:410d6c203b078554_8_86]] is countable but there is no correspondence between [[IMAGE:410d6c203b078554_8_87]] and [[IMAGE:410d6c203b078554_8_88]] .
[[IMAGE:410d6c203b078554_8_89]] is uncountable and therefore, there is no correspondence between [[IMAGE:410d6c203b078554_8_90]] and [[IMAGE:410d6c203b078554_8_91]]
.
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Question 13 MCQ · 3.0 marks
Which of the following is a formula representing the sequence of coefficients for the generating
[[IMAGE:410d6c203b078554_8_92]]
function ?
[[IMAGE:410d6c203b078554_8_93]]
[[IMAGE:410d6c203b078554_8_94]]
[[IMAGE:410d6c203b078554_8_95]]
[[IMAGE:410d6c203b078554_8_96]]
A published solution is not available for this question yet.
Question 14 MCQ · 3.0 marks
Let [[IMAGE:410d6c203b078554_8_97]] be the relation on the set of students consisting of pairs [[IMAGE:410d6c203b078554_8_98]] , where [[IMAGE:410d6c203b078554_8_99]] and [[IMAGE:410d6c203b078554_8_100]] are enrolled in
the same course. Let [[IMAGE:410d6c203b078554_8_101]] be the relation consisting of pairs [[IMAGE:410d6c203b078554_8_102]] , where [[IMAGE:410d6c203b078554_8_103]] and [[IMAGE:410d6c203b078554_8_104]] belong to the same
department. Which of the following correctly describes [[IMAGE:410d6c203b078554_8_105]] ?
The set of students who take the same course
The set of students who belong to the same department
The set of students [[IMAGE:410d6c203b078554_9_106]] and [[IMAGE:410d6c203b078554_9_107]] such that there exists a student [[IMAGE:410d6c203b078554_9_108]] who shares a
course with [[IMAGE:410d6c203b078554_9_109]] and a department with [[IMAGE:410d6c203b078554_9_110]] .
The set of students [[IMAGE:410d6c203b078554_9_111]] and [[IMAGE:410d6c203b078554_9_112]] such that there exists a student [[IMAGE:410d6c203b078554_9_113]] who shares a
department with [[IMAGE:410d6c203b078554_9_114]] and a course with [[IMAGE:410d6c203b078554_9_115]] .
Students who have identical course schedules
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Question 15 MCQ · 3.0 marks
Let [[IMAGE:410d6c203b078554_9_116]] and [[IMAGE:410d6c203b078554_9_117]] be relations on a set [[IMAGE:410d6c203b078554_9_118]] represented by the matrices
[[IMAGE:410d6c203b078554_9_119]]
Which of the following matrices represent [[IMAGE:410d6c203b078554_9_120]] ?
[[IMAGE:410d6c203b078554_9_121]]
[[IMAGE:410d6c203b078554_9_122]]
[[IMAGE:410d6c203b078554_9_123]]
None of these
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Question 16 NAT · 4.0 marks
How many positive integers between [[IMAGE:410d6c203b078554_10_124]] and [[IMAGE:410d6c203b078554_10_125]] inclusive are divisible by [[IMAGE:410d6c203b078554_10_126]] or [[IMAGE:410d6c203b078554_10_127]] ?
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Question 17 NAT · 2.0 marks
[[IMAGE:410d6c203b078554_10_128]]
[[IMAGE:410d6c203b078554_10_129]]
[[IMAGE:410d6c203b078554_10_132]]
If and for some value of [[IMAGE:410d6c203b078554_10_130]] and [[IMAGE:410d6c203b078554_10_131]] , then what is the value of ?
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Question 18 MSQ · 2.0 marks
[[IMAGE:410d6c203b078554_10_133]]
Reflexive Closure
Symmetric Closure
Transitive Closure
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