Description

Show all work needed to reach your answers.
1. (10 points) If A = {2,4,8}, then the power set of A is

2. (20 points) 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. (10 points) 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. (20 points) Please show that 5 is irrational.
5. (10 points) 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. (20 points) 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. (10 points) 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.