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Questions:

1. Show that the set of all bit strings is countable.
2. Prove that the set {0, 1, 2, 3, 4} is a finite abelian group of order 5 under addition modulo 5.
3. Let f(x) = x - 1 and g(x) = x^2 + 1, find (f g)(2) and (g g)(2).
4. When is a simple graph G bipartite? Give an example.
5. Show that the proposition p q and p q are logically equivalent.
6. Show that x(P(x) Q(x)) (xP(x)) (xQ(x)) by indirect method of proof.
7. Prove that 2 is irrational by giving a proof using contradiction.
8. Solve the recurrence relation: a_(n+1) - a_n = 3 2^n, n > 0, a_0 = 3.
9. Use mathematical induction to show a given statement.
10. State the pigeonhole principle and prove that if any 51 integers are chosen from the set {1, 2, ..., 100}, then there exist two integers such that one is a multiple of the other.
11. Prove that every finite group of order n is isomorphic to a permutation group of order n.
12. Prove that DeMorgan's laws hold good for a complemented distributive lattice.
13. Define with examples: Complete graph, Abelian group, Lattice, Platonic graph, Petersen graph, Biconditional statement, Isomorphism between two algebraic systems.