Homework 1, Due Thursday, 02/04

Questions 1, 2 & 3: Answer the three questions at the end of the Pirates, Pizzas and Preferences handout.

State whether each of the following relationships between sets is true or false. Justify your answer briefly. (Here, ∅ is the empty set.)

1. ∅ ∈ ∅

2. {∅} ∈ {∅}

3. ∅ ∈ {∅}

4. {∅} ⊆ {∅}

5. ∅ ⊆ ∅

Let R be the set of all real numbers and let A indicate the complement of the set A. We define the sets A, B and C as follows:
A = { xR | -3 ≤ x ≤ 0 }

B = { xR | -1 < x < 2 }

C = { xR | 6 < x ≤ 8 }

Describe the following sets:

1. AB

2. AC

3. AB

4. AB

5. AB

3. For each of the following functions, state whether the function is one-to-one, whether the function is onto and whether the function is a bijection. Pay close attention to the domain and codomain of each function. Briefly justify your answer.

1. f : NN,   f (n) = n2 + 5.

2. f : ZZ,   f (n) = n2 + 5.

3. f : NN,   f (n) = 2 n + 7.

4. f : RR,   f (x) = 2 x + 7.

5. f : RR,   f (x) = 2 x3 + 1.

Note: N, Z and R denote the set of natural numbers, integers and real numbers respectively.