2. Prove that (p (p q)) p q by using a series of logical equivalence.

3. Construct a truth table for each of these compound propositions. Also state the number of rows of each proposition using the formula 2n where n is the number of proposition variable.a) (p q) (p q) b) (q p) <-> (p q) c) (p q r) d) P p

4. Create the converse, contrapositive and inverse of the conditional statement: If you will work hard you will get success.

5. Write each of these English sentences in the form of proposition variable p, q with stating what is p and q in each case: (i) If you read the newspaper everyday then you will be informed. (ii) If you have connections you get promoted and conversely. (iii) You can see the wizard only if the wizard is not in, and the wizard is not in only if you can see him. (iv) For you to win the contest it is necessary and sufficient that you have the only winning ticket.

6. Let U = {1,2,3,4,5}, P = {3,4}, Q = {1,5,3}. Write down the elements of following sets. _ (i) (P U Q) P _ (ii) P U Q _ (iii) P _ (iv) Q

8. Let f and g be the functions from {1,2,3,4} to {a, b, c, d} and {a, b, c, d} to {1,2,3,4} respectively, with f(1) = d, f(2) =c, f(3) = a, f(4) = b and g(a)=2, g(b) = 1, g(c) = 3, g(d) =2. (i) Is f one to one? Is g one to one? (ii) Is f onto? Is g onto? (iii) Does either f or g have an inverse. If so find its inverse.

9. Determine whether each of these pairs of sets are equal with proper explanation: (i) {1,3,3,3,5,5,5} and {5,3,1} (ii) {1,2,3,3,4} and {1,2,3} (iii) and {} Also suppose that A = {2,4,6}, B = {2,6}, C={4,6} and D ={4,6,8}. Determine which of these are subset of which other of these sets.

10. Determine whether each of these functions is biject from R to R. a) f(x) = 2x+1 b) f(x) = -3x2 + 7 c) f(x) = x5 + 1 d) f(x) = (x2 + 1) / (x2 + 2)