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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Explain proof techniques: direct proof, proof by contraposition and proof by contradiction, with one example each. Prove that (\sqrt{2}) is irrational.

prooftechniques
2long10 marks

Define group, ring and field with examples. Show that the set of integers under addition forms a group. State and verify the group axioms.

algebraic-structuresgroup
3long10 marks

What is graph coloring? Explain the chromatic number of a graph. State the four color theorem. Find the chromatic number of a complete graph (K_n) and a cycle (C_n).

graphcoloring
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Define logical implication and biconditional.

logic
5short5 marks

Show that the function (f(x) = x^2) from R to R is not one-to-one.

functions
6short5 marks

Define Cartesian product of two sets with an example.

sets
7short5 marks

Use mathematical induction to prove (2^n > n) for (n \geq 1).

induction
8short5 marks

Define directed and undirected graphs with examples.

graph
9short5 marks

What is a minimum spanning tree? Name two algorithms to find it.

tree
10short5 marks

Define monoid and semigroup with examples.

algebraic-structures
11short5 marks

Solve (7x \equiv 1 \pmod{26}) to find the multiplicative inverse of 7.

number-theory
12short5 marks

State the binomial theorem and expand ((x+y)^4).

counting