Discrete Mathematics - End Semester Examination - 2023

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

1. 5
2. 9
3. 6
4. 4

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

1. \( m < n \)
2. \( m = n \)
3. \( m > n \)
4. none of these

If \( A \times B = B \times A \), (Where A and B are general matrices) then

1. \( A = \phi \)
2. \( A = B' \)
3. \( B = A \)
4. \( A' = B \)

A partial ordered relation is transitive, reflexive and

1. Anti symmetric
2. bi symmetric
3. anti reflexive
4. asymmetric

If B is a Boolean Algebra, then which of the following is true

1. B is a finite but complemented lattice
2. B is a finite, complemented and distributive lattice
3. B is a finite, Distributive but not complemented lattice
4. B is not distributive lattice

\( P \rightarrow q \) is logically equivalent to

1. \( -q \rightarrow p \)
2. \( -P \rightarrow q \)
3. \( -P \wedge q \)
4. \( -p \vee q \)

If \( f(x) = \cos x \) and \( g(x) = x^3 \) then \( (f \circ g)(x) \) is

1. \( (\cos x)^3 \)
2. \( \cos 3x \)
3. \( x^{(\cos x)^3} \)
4. \( \cos(x^3) \)

The number of distinguishable permutations of the letters in the word BANANA are

1. 60
2. 36
3. 20
4. 10

Which of the following pair is not congruent modulo 7?

1. 10, 24
2. 25, 56
3. -31, -15
4. -64, -15

Let \( N=\{1,2,3,........\} \) be ordered by divisibility, which of the following subset is totally ordered

1. (2, 6, 24)
2. (3, 5, 15)
3. (2, 9, 16)
4. (4, 15, 30)

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

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

Find the power set of each of these sets: i) \( \{a,b\} \) ii) \( \{\phi, \{\phi\}\} \)

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

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 \( f: Z_{+} \rightarrow \{0,1\} \)

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

State and prove Division algorithm theorem well-ordering principle.

Check the validity of the following argument all integers are rational numbers. Some integers are powers of 5. Therefore, some rational numbers are powers of 5.

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?

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.

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

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