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




There are 5 questions to complete.

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