The document outlines topics in mathematical theory including: defining functions and relations; properties of functions such as composition, inverses, one-to-one and onto functions; properties of relations including reflexivity, symmetry and transitivity; using principles of mathematical induction to prove statements; proving statements using proof by contradiction; and closure properties of regular languages. Specific problems include proving square root of 2 is irrational, using induction to prove a summation formula, and analyzing relations.

1. UNIT 1: REVIEW OF MATHEMATICAL THEORY
1. Define one-to-one, onto and bijection function.
2. Explain Compositions and Inverse of functions.
3. Explain reflexivity, symmetry, and transitivity properties of relations
4. Check whether the function f: R+  R, f(x) = x2 is one to one and onto.
5. Explain equivalence relation with example.
6. Using Principle of Mathematical Induction, prove that for every n >= 1.
7. Prove that √2 is Irrational by method of Contradiction.
8. Consider the relation R = {(1,2), (1,1), (2,1), (2,2), (3,2), (3,3)} defined over {1, 2,
3}. Is it reflexive? Symmetric? Transitive? Justify each of your answers.
9. Draw truth table for following logic formula: P  (¬P V ¬Q). Is it a tautology? A
contradiction? Or neither? Justify your answer.
10. Use the principle of mathematical induction to prove that 1 +3 +5 + … +r = n2 for
all n>0 where r is an odd integer & n is the number of terms in the sum. (Note: r=
2n-1)
11. Let A = {1, 2, 3, 4, 5, 6} and R be a relation on A such that aRb iff a is a multiple of
b. Write R. Check if the relation is i) Reflexive ii) Symmetric iii) Asymmetric
iv) Transitive.
12. Define Mathematical Induction Principle and Prove that for every n ≥ 1,
13. Prove the formula (00*1)*1 = 1+0(0+10)*11
14. State the principle of mathematical induction and prove by mathematical
induction that for all positive integers n 1+2+3+……. +n = n (n+1)/2.
15. What are the closure properties of regular languages?
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