MA1971 Bridge to Higher Math Name:
Final D Term,
Show all work needed to reach your answers.
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:
(a) contrapositive:
(b) converse:
(c) negation:
(d) 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 < 0.” Please avoid the use of the words “not” and “no”.
(a) Contrapositive:
(b) Negation:
√
4.Please show that 5 is irrational.
5.Consider a sequence {an} where an = pn/qn and 0 < pn < qn (so each element of the sequence is a fraction). Suppose that an is increasing. Does {an} 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.
