HW #1 due Monday Oct 2

1. Consider the following propositions about a function f(x,y) of two real numbers:

P1. for all x and y, f(x,y)=0

P2. for all x, there exists a y such that f(x,y)=0

P3. for all y, there exists an x such that f(x,y)=0

P4. there exists an x such that for all y, f(x,y)=0

P5. there exists a y such that for all x, f(x,y)=0

P6. there exist x and y such that f(x,y)=0.

a. For the following functions f, which of P1-P6 are true?

i. f(x,y) = 0 everywhere. ANSWER: "P1,P2,P3,P4,P5,P6 are all true."
ii. f(x,y) = x+y.
iii. f(x,y) = x³-xy²
iv. f(x,y) = y-x².
v. f(x,y) = x+y².
vi. f(x,y) = x²+y²-1

b. How do things change in part (a) if we allow x,y to be complex numbers?

2. Consider the following propositions about a function f(x,y,z) of three real numbers:

P1. for all x, there exists a y such that for all z, f(x,y,z)=0

P2. there exists an x such that for all y, there exists a z such that f(x,y,z)=0

a. Write down a function f(x,y,z) that satisfies P1 but not P2. Explain how to find the y required in P1, and why the x in P2 doesn't exist.

b. Write down a function f(x,y,z) that satisfies P2 but not P1. Explain how to find the x and z required in P2, and in P1 give an x for which there is no good y.

Warning: the reason this question is hard is that it has a bajillion possible answers, all equally good. Answering this question is more art than science.

3. Definition. A graph G is vague-ular if for all {v,w} in E_G, degree(v) = degree(w).

a. Prove that if a graph G is regular, it is vague-ular.

b. Show that the converse, "if a graph G is vague-ular, it is regular" is not true.

(What does this mean, "show this implication is not true"? It means give an explicit example of a graph that is vague-ular but not regular.)

The rest are from [FP], meaning, Fletcher and Patty. They're not as hard -- more just to help you check that you know what's going on.

1.1 #2, 3, 4, 8, 11
1.2 #21 (true, false, sometimes), 27
1.3 #52 (write these out with quantifiers)