2023 question paper

Discrete Mathematics

19 questions

1. 1a. Let A be the set odd positive integers less than 10. Then cardinality of A, ∣A∣ is (i) 5 (ii) 9 (iii) 6 (iv) 420232m

   Module 1: Sets, Relation and Function.

   Let A be the set odd positive integers less than 10. Then cardinality of A, ∣A∣ is

   (i) 5
   (ii) 9
   (iii) 6
   (iv) 4

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   Worked Solution

   Answer

   Correct Option: (i) 5

   Explanation

   $A = \{ x \in \mathbb{Z}^+ \mid x \text{ is odd and } x < 10 \}$

   Listing elements:
   $$A = \{1, 3, 5, 7, 9\}$$

   Cardinality $|A|$ is the number of distinct elements in $A$.

   Counting:
   $$|A| = 5$$

   Final Answer

   $$|A| = 5$$ → Option (i) 5

2. 1b. If m is the number of objects (pigeons) and n is the number of boxes (pigeonholes), then the function is both one-to-one and onto if (i) m<n (ii) m=n (iii) m>n (iv) none of these20232m

   Module 1: Sets, Relation and Function.

   If m is the number of objects (pigeons) and n is the number of boxes (pigeonholes), then the function is both one-to-one and onto if
   (i) m<n
   (ii) m=n
   (iii) m>n
   (iv) none of these

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3. 1c. If A \times B = B \times A, (Where A and B are general matrices) then (i) A=I (ii) A=B' (iii) B=A (iv) A'=B20232m

   Module 4: Algebraic Structures & Boolean Algebra.

   If $A \times B = B \times A$, (Where A and B are general matrices) then
   (i) $A=I$
   (ii) $A=B'$
   (iii) $B=A$
   (iv) $A'=B$

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4. 1g. If f(x) = \cos x and g(x) = x^3 then (f \circ g)(x) is (i) (\cos x)^3 (ii) \cos 3x (iii) x^{(\cos x)^3} (iv) \cos x^320232m

   Module 1: Sets, Relation and Function.

   If $f(x) = \cos x$ and $g(x) = x^3$ then $(f \circ g)(x)$ is
   (i) $(\cos x)^3$
   (ii) $\cos 3x$
   (iii) $x^{(\cos x)^3}$
   (iv) $\cos x^3$

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5. 1h. The number of distinguishable permutations of the letters in the word BANANA are (i) 60 (ii) 36 (iii) 20 (iv) 1020232m

   Module 1: Sets, Relation and Function.

   The number of distinguishable permutations of the letters in the word BANANA are
   (i) 60
   (ii) 36
   (iii) 20
   (iv) 10

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6. 1i. Which of the following pair is not congruent modulo 7? (i) 10, 24 (ii) 25, 56 (iii) -31, -15 (iv) -64, -1520232m

   Module 4: Algebraic Structures & Boolean Algebra.

   Which of the following pair is not congruent modulo 7?
   (i) 10, 24
   (ii) 25, 56
   (iii) -31, -15
   (iv) -64, -15

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7. 1j. Let N = \{1,2,3,......\} be ordered by divisibility, which of the following subset is totally ordered (i) \{2, 6, 24\} (ii) \{3, 5, 15\} (iii) \{2, 9, 16\} (iv) \{4, 15, 30\}20232m

   Module 1: Sets, Relation and Function.

   Let $N = \{1,2,3,......\}$ be ordered by divisibility, which of the following subset is totally ordered
   (i) $\{2, 6, 24\}$
   (ii) $\{3, 5, 15\}$
   (iii) $\{2, 9, 16\}$
   (iv) $\{4, 15, 30\}$

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8. 2a. a) Let A = B = \{x | -1 \leq x \leq 1\} for each of the following functions state where it is injective, surjective or bijective i) g(x) = \sin \pi x ii) h(x) = \frac{2x}{3}20237m

   Module 1: Sets, Relation and Function.

   a) Let $A = B = \{x | -1 \leq x \leq 1\}$ for each of the following functions state where it is injective, surjective or bijective
   i) $g(x) = \sin \pi x$
   ii) $h(x) = \frac{2x}{3}$

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9. 2b. Use Cantor's diagonal argument to prove that set F of all functions f: (0,1) \rightarrow R has larger Cardinality than |R|.20237m

   Module 1: Sets, Relation and Function.

   Use Cantor's diagonal argument to prove that set F of all functions $f: (0,1) \rightarrow R$ has larger Cardinality than $|R|$.

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10. 2b. Let f(x) = x+2, g(x) = x-2, h(x) = 3x find (i) f \circ g (ii) f \circ g \circ h20237m

    Module 1: Sets, Relation and Function.

    Let $f(x) = x+2$, $g(x) = x-2$, $h(x) = 3x$ find
    (i) $f \circ g$
    (ii) $f \circ g \circ h$

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12. 3a. a.) find the power set of each of these sets i) \{a, b\} ii) \{\emptyset, \{\emptyset\}\}20237m

    Module 1: Sets, Relation and Function.

    a.) find the power set of each of these sets
    i) $\{a, b\}$
    ii) $\{\emptyset, \{\emptyset\}\}$

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13. 4. Determine if the sets are countable or uncountable a.) the set A of all function g: Z_+ \rightarrow Z_+ b.) The set B of all functions f: Z_+ \rightarrow \{0,1\}202314m

    Module 1: Sets, Relation and Function.

    Determine if the sets are countable or uncountable
    a.) the set A of all function $g: Z_+ \rightarrow Z_+$
    b.) The set B of all functions $f: Z_+ \rightarrow \{0,1\}$

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14. 5. Prove the following by using the principle of mathematical induction for all n \in N: 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2202314m

    Module 2: Mathematical Induction & Counting Techniques.

    Prove the following by using the principle of mathematical induction for all $n \in N$:
    $$1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$

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    Worked Solution

    Answer

    Prove: $1^3 + 2^3 + \dots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$ for all $n \in \mathbb{N}$ by Mathematical Induction.

    Step 1: Base Case ($n=1$)

    LHS $= 1^3 = 1$

    RHS $= \left(\dfrac{1\cdot 2}{2}\right)^2 = 1^2 = 1$

    LHS = RHS, so holds for $n=1$.

    Step 2: Induction Hypothesis

    Assume true for $n=k$:
    $$P(k): 1^3 + 2^3 + \dots + k^3 = \left(\frac{k(k+1)}{2}\right)^2$$

    Step 3: Inductive Step ($n = k+1$)

    To prove $P(k+1)$:
    $$1^3 + \dots + k^3 + (k+1)^3 = \left(\frac{(k+1)(k+2)}{2}\right)^2$$

    Consider LHS of $P(k+1)$:
    $$\begin{aligned} \text{LHS} &= \left(1^3 + \dots + k^3\right) + (k+1)^3 \\ &= \left(\frac{k(k+1)}{2}\right)^2 + (k+1)^3 \quad \text{(by hypothesis)}\\ &= \frac{k^2(k+1)^2}{4} + (k+1)^3 \\ &= (k+1)^2 \left[ \frac{k^2}{4} + (k+1) \right] \\ &= (k+1)^2 \left[ \frac{k^2 + 4k +4}{4} \right] \\ &= (k+1)^2 \cdot \frac{(k+2)^2}{4} \\ &= \left( \frac{(k+1)(k+2)}{2} \right)^2 = \text{RHS of } P(k+1) \end{aligned}$$

    Thus $P(k) \implies P(k+1)$.

    Step 4: Conclusion

    By the Principle of Mathematical Induction, since base case holds and inductive step proved, the formula holds for all $n \in \mathbb{N}$.

    Corollary / Remark

    Note that $1+2+\dots+n = n(n+1)/2$, so the theorem elegantly states:
    $$1^3 + 2^3 + \dots + n^3 = (1+2+\dots+n)^2$$

15. 6. State and prove Division algorithm theorem well-ordering principle.202314m

    Module 2: Mathematical Induction & Counting Techniques.

    State and prove Division algorithm theorem well-ordering principle.

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16. 7a. Check the validity of the following argument all integers are rational numbers. Some integers are powers of 5. There fore, some rational numbers are powers of 520237m

    Module 3: Propositional Logic & Proof Techniques.

    Check the validity of the following argument all integers are rational numbers. Some integers are

    powers of 5. There fore, some rational numbers are powers of 5

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17. 7b. A grocery store employee is stocking apples. Each apple is a different color. There are 10 apples left in the box and the employee pulls out 2 of them at random. What is the probability that the employee pulls out one pink apple and yellow apple?20237m

    Module 2: Mathematical Induction & Counting Techniques.

    A grocery store employee is stocking apples. Each apple is a different color. There are 10 apples left

    in the box and the employee pulls out 2 of them at random. What is the probability that the employee

    pulls out one pink apple and yellow apple?

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18. 8. Let \Psi : G \rightarrow H be a homomorphism of groups. Show that if a \in G has order n, then \Psi (a) \in H has order dividing n.202314m

    Module 4: Algebraic Structures & Boolean Algebra.

    Let $\Psi : G \rightarrow H$ be a homomorphism of groups. Show that if $a \in G$ has order $n$, then $\Psi (a) \in H$ has order dividing $n$.

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19. 9a. Consider the following graph (a) Does a Hamiltonian path exist? If so describe it. If not say why not.20237m

    Module 5: Graphs and Trees

    Consider the following graph

    (a) Does a Hamiltonian path exist? If so describe it. If not say why not.

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20. 9b. Does an Eulerian path exist? If so describe it. If not say why not.20237m

    Module 5: Graphs and Trees

    Does an Eulerian path exist? If so describe it. If not say why not.

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