Introduction: Sets And Functions

Question 1

Define what is meant by set

Answer 1

A set is a collection of distinct objects, considered as a unit.

Question 2

What is “Z” notation?

Answer 2

The set of integers {…-3, -2, -1, 0, 1, 2, 3…}

Question 3

What is “N” notation?

Answer 3

The set of natural numbers {1, 2, 3…}

To include 0: N subscript 0

Question 4

What is “Q” notation?

Answer 4

The set of rational numbers {a/b: a,b € Z, b#0}

To include positive numbers do superscript + (0 is not positive)

Question 5

What is “R” notation?

Answer 5

The set of real numbers (including surds)

To include positive numbers do superscript + (0 is not positive)

Question 6

What is “C” notation?

Answer 6

The set of complex numbers C = {a+ib: a,b € R}

Question 7

What is “{}” or “Ø” notation?

Answer 7

The empty set, with no elements

Question 8

Define what is meant by subset

Answer 8

A set a is a subset of a set X, denoted by A C [underlined] X, if every element of A is also an element of X. We write A C/ [underlined and through] B when A is not a subset of B

Question 9

Define what is meant by proper subset

Answer 9

A set A is a proper subset of a set X, denoted by A C X, if A is a subset of X but A#X

Question 10

What notation is the only subset of itself?

Answer 10

Ø

Question 11

What is the order of subsets R, C, Z and Q?

Answer 11

Z C [underlined] Q C [underlined] R C [underlined] C

Question 12

How do you show A=X using sets?

Answer 12

A C [underlined] X and X C [underlined] A

Question 13

Define what is meant by linearly ordered

Answer 13

A set X which is linearly ordered satisfies the following properties. For x,y,z € X
Either x=y or y=x
If x=y and y=x then x=y
If x=y and y=z then x=z
Any subset of the real numbers are linearly ordered

Question 14

Define what is meant by bounded above

Answer 14

We say a (non empty) subset A of R is bounded above is there exists some M€R such that a=M, for all a€A. Then M contains a maximal element.
If there is a (maximal) value greater than or equal to every number in the range.

Question 15

Define what is meant by bounded below

Answer 15

We say that A is bound below is there is some N€R such that N=a. For all a€A. Then N contains a minimal element.
If there is a (minimal) value less than or equal to every number in the range.

Question 16

Define what is meant by Cartesian product

Answer 16

Given the 2 sets X and Y, the Cartesian product is denoted by X*Y ={(x,y) : x€X, y€Y}
if X or Y = Ø then the product = Ø

Question 17

Define what is meant by function

Answer 17

Let X, Y be non empty sets a function from X into Y f:X–>Y is a set of ordered pairs f C [underlined] X*Y with the property that for each element x€X there exists one element f(x) € Y such that (x, f(x)) € f