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Questions:

1. Show that the mapping f : R -> R defined by f(x) = 3x + 4, where x R, is invertible. Find its inverse.
2. Let X = {1, 2, 3, 4, 5, 6}. Then f is a partial order relation on X. Draw the Hasse diagram of (X, f).
3. Show that the permutation (1 2 3 4 5 6) (5 6 2 4 1 3) is odd.
4. Construct a truth table for the proposition (p q) p.
5. Define simple graph and multigraph with examples.
6. If A and B are two sets, then prove that (A B)' = A' B'.
7. Explain graph coloring.
8. Let G be a finite group and H be a subgroup of G. Show that |H| divides |G|.
9. State and prove the pigeonhole principle.
10. Show that n > 2n + 1 for n 3 by mathematical induction.
11. Define a group. Prove that the fourth roots of unity {1, -1, i, -i} form an abelian group under multiplication.
12. Use Karnaugh map to simplify the expression: X = A'B'C'D' + AB'C'D' + A'B'CD' + AB'CD'.
13. Let A = {1, 2, 3, 4} and consider the relation R = {(1, 1), (2, 1), (2, 2), (3, 1), (3, 3), (3, 4), (4, 4)}. Show that R is a partial ordering.
14. Solve the recurrence relation: a - 5a + 6a = 2 with initial conditions a = 1, a = -1.
15. Explain the basic logical operations of propositional calculus.