[Solved] MA1971 Final Exams

1. If A = {2,4,8}, then the power set of A is
2. Consider the implication A B where A and B are themselves statements or predicates. For this implication, please state the following:
   • contrapositive:
   • converse:
   • negation:
   • inverse:
3. Please give (a) the contrapositive and then (b) the negation of the following statement: If xy is an irrational number, then y > 6 but x < Please avoid the use of the words not and no.
   • Contrapositive:
   • Negation:

4. Please show that 5 is irrational.
5. Consider a sequence {a_n} where a_n = p_n/q_n and 0 < p_n < q_n (so each element of the sequence is a fraction). Suppose that a_n is increasing. Does {a_n} necessarily converge? Please either explain why it converges, or give a counterexample to show that such a sequence might diverge.
6. Please explain why i² +j² is never equal to 3 (mod 4), that is, i² +j² 6= 4k +3 for any i, j, k Z.

Hint: Consider the cases where i and j are each either even or odd; what do these imply?

7. Consider the graph below; it is one possible drawing of K₅, the complete graph on five vertices. Recall that by the Euler formula, one might expect that |V ||E|+|F| = 2. But for this graph, it seems that |V | = 5, |E| = 10 and |F| = 8, meaning that the Euler formula is not satisfied. Please explain what is wrong here.