Discrete Mathematics

Question 1
Let G be a simple, finite, undirected graph with vertex set \{v_1,...,v_n\}. Let \Delta (G) denote the maximum degree of G and let N=\{1,2,...\} denote the set of all possible colors. Color the vertices of G using the following greedy strategy: for i = 1,...,n
color(v_i)\leftarrow min\{ j \in N: \text{ no neighbour of } v_i \text{ is colored } j\}
Which of the following statements is/are TRUE?
A
This procedure results in a proper vertex coloring of G.
B
The number of colors used is at most \Delta (G)+1.
C
The number of colors used is at most \Delta (G).
D
The number of colors used is equal to the chromatic number of G.
GATE CSE 2023      Graph Theory
Question 2
Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?
A
A and B are independent.
B
A and C are independent.
C
B and C are independent.
D
Prob(B|C) = Prob(B)
GATE CSE 2023      Probability Theory
Question 3
Let X be a set and 2^X denote the powerset of X.
Define a binary operation \Delta on 2^X as follows:
A \Delta B=(A-B) \cup (B-A)
Let H=(2^X,\Delta ) . Which of the following statements about H is/are correct?
A
H is a group.
B
Every element in H has an inverse, but H is NOT a group.
C
For every A \in 2^X, the inverse of A is the complement of A.
D
For every A \in 2^X, the inverse of A is A.
GATE CSE 2023      Set Theory
Question 4
Let f:A\rightarrow B be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation \sim on the set A as
a_1\sim a_2 \text{ if } f(a_1)=f(a_2),
where a_1, a_2 \in A . Let \varepsilon =\{[x]:x \in A\} be the set of all the equivalence classes under \sim . Define a new mapping F: \varepsilon \rightarrow B as
F([x]) = f(x), for all the equivalence classes [x] in \varepsilon
Which of the following statements is/are TRUE?
A
F is NOT well-defined.
B
F is an onto (or surjective) function.
C
F is a one-to-one (or injective) function.
D
F is a bijective function.
GATE CSE 2023      Function
Question 5
Let U = {1, 2,...,n}, where n is a large positive integer greater than 1000. Let k be a positive integer less than n. Let A, B be subsets of U with |A| = |B| = k and A\cap B=\phi . We say that a permutation of U separates A from B if one of the following is true.

- All members of A appear in the permutation before any of the members of B.
- All members of B appear in the permutation before any of the members of A.

How many permutations of U separate A from B?
A
n!
B
\binom{n}{2k} (n-2k)!
C
\binom{n}{2k} (n-2k)! (k!)^2
D
2 \binom{n}{2k} (n-2k)! (k!)^2
GATE CSE 2023      Combination
Question 6
Geetha has a conjecture about integers, which is of the form
\forall x\left [P(x)\Rightarrow \exists yQ(x,y) \right ]
where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?
A
\exists x\left [P(x)\wedge \forall yQ(x,y) \right ]
B
\forall x \forall y Q(x,y)
C
\exists y \forall x \left [P(x) \Rightarrow Q(x,y) \right ]
D
\exists x \left [P(x) \wedge \exists y Q(x,y) \right ]
GATE CSE 2023      Propositional Logic
Question 7
The Lucas sequence L_n is defined by the recurrence relation:
L_n=L_{n-1}+L_{n+2}, \; for \; n\geq 3,
with L_1=1 \; and \; L_2=3
Which one of the options given is TRUE?
A
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n
B
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n - \left ( \frac{1-\sqrt{5}}{3} \right )^n
C
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{3} \right )^n
D
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n- \left ( \frac{1-\sqrt{5}}{2} \right )^n
GATE CSE 2023      Recurrence
Question 8
Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?
MSQ
A
The diagonal entries of A^2 are the degrees of the vertices of the graph.
B
If the graph is connected, then none of the entries of A^{n-1}+I_ncan be zero.
C
If the sum of all the elements of A is at most 2(n-1), then the graph must be acyclic.
D
If there is at least a 1 in each of A's rows and columns, then the graph must be connected.
GATE CSE 2022      Graph Theory
Question 9
Consider the following recurrence:
\begin{aligned} f(1)&=1; \\ f(2n)&=2f(n)-1, & \text{for }n \geq 1; \\ f(2n+1)&=2f(n)+1, & \text{for }n \geq 1. \end{aligned}
Then, which of the following statements is/are TRUE?
MSQ
A
f(2^n-1)=2^n-1
B
f(2^n)=1
C
f(5 \dot 2^n)=2^{n+1}+1
D
f(2^n+1)=2^n+1
GATE CSE 2022      Recurrence
Question 10
The following simple undirected graph is referred to as the Peterson graph.

Which of the following statements is/are TRUE?
MSQ
A
The chromatic number of the graph is 3.
B
The graph has a Hamiltonian path.
C
The following graph is isomorphic to the Peterson graph.

D
The size of the largest independent set of the given graph is 3. (A subset of vertices of a graph form an independent set if no two vertices of the subset are adjacent.)
GATE CSE 2022      Graph Theory


There are 10 questions to complete.

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