Math Problems Vault

Problem 0.1

Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set.

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(a)

$\left\{ 1, 3, 5, 7, \ldots \right\}$

(b)

$\left\{ \ldots ,-4,-2,0,2,4,\ldots \right\}$

(c)

$\left\{ n \mid n=2 m\text{ for some }m\text{ in }\mathcal{N}\right\}$

(d)

$\left\{ n \mid n=2 m\text{ for some }m\text{ in }\mathcal{N}\text{, and }n=3 k\text{ for some }k\text{ in }\mathcal{N}\right\}$

(e)

$\left\{ w \mid w\text{ is a string of 0s and 1s and }w\text{ equals the reverse of }w\right\}$

(f)

$\left\{ n \mid n\text{ is an integer and }n=n+1\right\}$

Problem 0.2

Write formal descriptions of the following sets.

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(a)

The set containing the numbers 1, 10, and 100

(b)

The set containing all integers that are greater than 5

(c)

The set containing all natural numbers that are less than 5

(d)

The set containing the string aba

(e)

The set containing the empty string

(f)

The set containing nothing at all

Problem 0.3

Let $A$ be the set $\left\{ \mathrm{x}, \mathrm{y}, \mathrm{z}\right\}$ and $B$ be the set $\left\{ \mathrm{x}, \mathrm{y}\right\}$ .

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(a)

Is $A$ a subset of $B$ ?

(b)

Is $B$ a subset of $A$ ?

(c)

What is $A \cup B$ ?

(d)

What is $A \cap B$ ?

(e)

What is $A \times B$ ?

(f)

What is the power set of $B$ ?

Problem 0.4

If $A$ has $a$ elements and $B$ has $b$ elements, how many elements are in $A \times B$ ? Explain your answer.

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Problem 0.5

If $C$ is a set with $c$ elements, how many elements are in the power set of $C$ ? Explain your answer.

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Problem 0.6

Let $X$ be the set $\left\{ 1,2,3,4,5\right\}$ and $Y$ be the set $\left\{ 6,7,8,9,10\right\}$ . The unary function $f: X \longrightarrow Y$ and the binary function $g: X \times Y \longrightarrow Y$ are described in the following tables.

$n$	$f(n)$
1	6
2	7
3	6
4	7
5	6

$g$	6	7	8	9	10
1	10	10	10	10	10
2	7	8	9	10	6
3	7	7	8	8	9
4	9	8	7	6	10
5	6	6	6	6	6

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(a)

What is the value of $f(2)$ ?

(b)

What are the range and domain of $f$ ?

(c)

What is the value of $g(2,10)$ ?

(d)

What are the range and domain of $g$ ?

(e)

What is the value of $g(4, f(4))$ ?

Problem 0.7

(a)(b)(c)

For each part, give a relation that satisfies the condition.

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(a)

Reflexive and symmetric but not transitive

(b)

Reflexive and transitive but not symmetric

(c)

Symmetric and transitive but not reflexive

Problem 0.8

Consider the undirected graph $G=(V, E)$ where $V$ , the set of nodes, is $\left\{ 1,2,3,4\right\}$ and $E$ , the set of edges, is $\left\{ \left\{ 1,2\right\} ,\left\{ 2,3\right\} ,\left\{ 1,3\right\} ,\left\{ 2,4\right\} ,\left\{ 1,4\right\} \right\}$ . Draw the graph $G$ . What are the degrees of each node? Indicate a path from node 3 to node 4 on your drawing of $G$ .

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Problem 0.9

Write a formal description of the following graph.

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