Discrete Mathematics - B. Tech 4th Semester Examination, 2024

The function \( f:R \rightarrow R \) defined by \( f(x)=x^3+5 \)

1. One-One but not onto
2. Onto but not One-One
3. Both One-One and Onto
4. None of these

Consider the following relation R on a set \( A=\{1,2,3,4\} \): R = {(1,1), (1, 3), (3, 2), (2, 2), (3, 3), (3, 1), (2, 3), (1, 4), (4, 4)} Then

1. Reflexive
2. Symmetric
3. Transitive
4. Equivalence

Out of the following which of these integers is not prime?

1. 23
2. 59
3. 21
4. 71

The Greatest common divisor of 0 and 11

1. 0
2. 1
3. 11
4. Does not exist

Suppose G is a group with a binary operation * defined by \( a*b=a+b+1 \), for \( a, b \in G \) then

1. Identity element = -1
2. Identity element = 1
3. Identity element = -2
4. Identity element = 2

The field which contains at least _____ element/elements

1. 0
2. 1
3. 2
4. 3

The set of integers with respect to addition is

1. Abelian group
2. Cyclic group
3. Both (i) and (ii)
4. None of these

The total number of edges in a complete graph of n vertices is

1. \( n \)
2. \( \frac{n}{2} \)
3. \( n^2-1 \)
4. \( \frac{n(n-1)}{2} \)

In any undirected graph, the sum of degrees of all vertices

1. must be even
2. is a twice the number of edges
3. must be odd
4. Both (i) and (ii)

\( (p \rightarrow q) \wedge (r \rightarrow q) \) is equivalent to

1. \( (p \vee r) \rightarrow q \)
2. \( p \vee (r \rightarrow p) \)
3. \( p \vee (r \rightarrow q) \)
4. \( p \rightarrow (q \rightarrow r) \)

Given \( A=\{1,2,3\} \), \( B=\{7,8\} \) and \( R=\{(1,7), (2, 7), (1, 8), (3, 8)\} \), find \( R^{-1} \) (inverse of R) and \( R' \) complement of R.

Consider \( A=B=C=R \) set of real numbers and let \( f:A \rightarrow B \), \( g: B \rightarrow C \) defined by \( f(x)=x+9 \) and \( g(y)=y^2+3 \) find the following composition functions: \( (f \circ g)(a) \), \( (g \circ f)(a) \), \( (g \circ f)(x) \), \( (f \circ g)(-3) \).

What is the negation of each of the following propositions? (i) Today is Tuesday (ii) A cow is an animal (iii) No one wants to buy my house

Determine the truth value of each of the following statements (i) \( 4+3=7 \) and \( 6+2=8 \) (ii) \( 2+3=4 \) and \( 3+1=2 \)

Show that the set of rational numbers Q forms an abelian group.

Define a ring, give an example of a commutative ring without zero divisors and a non-commutative ring with identity.

Describe the Fundamental Theorem of Arithmetic. Factorize the number '324' and represent it as a product of primes.

Use the Euclidean algorithm to find the greatest common divisor of each pair of integers, (i) 60, 90 (ii) 414, 662

Describe a graph model that can be used to represent all forms of electronic communication between two people in a single graph. What kind of graph is needed?

Explain the complete graph and prove that \( k_5 \) (complete graph) is a nonplanar graph.

Explain the congruence relation. Let \( n \) be a positive integer then: (i) If \( a \equiv b \pmod{n} \), then \( b \equiv a \pmod{n} \) (ii) If \( a \equiv b \pmod{n} \) and \( b \equiv c \pmod{n} \), then \( a \equiv c \pmod{n} \).

Show that \( p \Leftrightarrow q \equiv (p \Rightarrow q) \wedge (q \Rightarrow p) \)

Write short notes of the following: (i) Eulerian Graph (ii) Graph Coloring

Write short notes of the following: (i) Chromatic number (ii) Perfect Graph

What is the field? Show that \( (Q, +, \times) \) is a field.

If in a ring R with \( (xy)^2=x^2y^2 \) for all \( x, y \in R \), then R is commutative.