UMBC CMSC203 Discrete Structures, Section 06, Spring 2016

Homework 1, Due Thursday, 02/04

4. [Adapted from Rosen 6/e]
   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. ∅ ⊆ ∅

5. [Adapted from Epp 3/e]
   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 = { x ∈ R | -3 ≤ x ≤ 0 }

   B = { x ∈ R | -1 < x < 2 }

   C = { x ∈ R | 6 < x ≤ 8 }

   Describe the following sets:

   1. A ∪ B

   2. A ∩ C

   3. A ∪ B

   4. A ∩ B

   5. A ∪ B

6. 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 : N → N,   f (n) = n² + 5.

   2. f : Z → Z,   f (n) = n² + 5.

   3. f : N → N,   f (n) = 2 n + 7.

   4. f : R → R,   f (x) = 2 x + 7.

   5. f : R → R,   f (x) = 2 x³ + 1.

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