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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define quantifiers. Translate the following into logical expressions using predicates and quantifiers, and find their negations: (a) Every student in this class has studied calculus. (b) Some student in this class owns a computer.

quantifierspredicate-logic
2long10 marks

State and explain the pigeonhole principle. How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? Also solve a generalized example.

countingpigeonhole
3long10 marks

Define relation. Explain reflexive, symmetric, antisymmetric and transitive relations with examples. What is an equivalence relation? Show that congruence modulo m is an equivalence relation.

relationsequivalence
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

State De Morgan's laws for sets and for logic.

logicsets
5short5 marks

Define onto and one-to-one functions.

functions
6short5 marks

Solve (a_n = a_{n-1} + 2n), (a_0 = 1) for (a_3).

recurrence
7short5 marks

What is the degree of a vertex? State the handshaking theorem.

graph
8short5 marks

Define a subgroup with an example.

algebraic-structures
9short5 marks

Differentiate between a tree and a graph.

treegraph
10short5 marks

Write the contrapositive, converse and inverse of 'If it rains then the ground is wet'.

logic
11short5 marks

How many bit strings of length 8 contain exactly three 1's?

counting
12short5 marks

Define adjacency matrix of a directed graph with an example.

graph