Discrete Math - Speaking Mathematically

Question 1

What is Mathematical Thinking?

Answer 1

Thinking in terms of a mathematical language that expresses mathematical ideas clearly, precisely, and unambiguously.

Thinking in terms of a mathematical language that is a foundation for much mathematical thought; the language of variables, sets, relations, and functions.

Question 2

What are the four types of mathematical thought in mathematical thinking?

Answer 2

1. Variables
2. Sets
3. Relations
4. Functions

Question 3

What is a Variable in Mathematical Thinking?

Answer 3

A variable is a placeholder for a value.

Question 4

What is a Set in Mathematical Thinking?

Answer 4

A set is a collection of elements.

Question 5

What is a Relation in Mathematical Thinking?

Answer 5

A relation is a comparison between mathematical objects.

Question 6

What is a Function in Mathematical Thinking?

Answer 6

A function takes a mathematical object as input, performs mathematics that input and produces a resulting output mathematical object.

Question 7

What are two ways that variables are useful?

Answer 7

1. Imagining that a mathematical formula has one or more values but they are not known, so a variable (i.e. placeholder) is used.2x + 3 = x2 where x is the variable
2. whatever you say about a mathematical formula to be equally true for all elements in a given set, and so you don’t want to be restricted to considering only a particular, concrete value for that mathematical formula.No matter what number n might be chosen,
   if n is greater than 2, then n2 is greater than 4.

Question 8

In set notation, what does ‘x ∈ S’ mean?

Answer 8

‘x ∈ S’ means that x is an elements of set S.

Question 9

In set notation, what does ‘x /∈ S’ mean?

Answer 9

‘x /∈ S’ mean that x is NOT and element of set S.

Question 10

In set notation, what is the ‘set-roster notation’?

Answer 10

The set-roster notation is a list of set elements between curly braces:
{1, 2, 3} where 1, 2 and 3 are elements of a set.

Question 11

What does the ‘axiom of extension’ say about a set?

Answer 11

The ‘axiom of extension’ says that a set is completely determined by what its elements are.
. The order of the elements listed doesn’t determine a set.
. repeated elements listed doesn’t determine a set.

Question 12

Describe the common sets: R, Z, Q and N.

Answer 12

R - set of all real numbers
Z - set of all integers
Q - set of all rational numbers, or quotients of integers
N - natural numbers

Question 13

Describe some superscripts for sets.

Answer 13

+ denotes a set of positive numbers. R+ demotes a set of positive real numbers

• denotes a set of negative numbers.

nonneg denotes a set of nonnegative numbers. Znonneg refers to the set of nonnegative integers.