Introduction

This handout is a summary of the basic concepts you should understand and be comfort- able to be able to successfully pass MATA31 course. This is intended as a summary and should be used together with Chapter 1 of your text book. Some parts in the notes are left uncompleted deliberately . You need to attend in the lecture and try to complete the notes on your own.

Sets

A set is a collection of objects. The objects of a set are called elements or members.

The set of natural numbers:

◮ are needed for counting The set of integers:

N = {1, 2, 3, . . .}

Z = {. . . − 2, −1, 0, 1, 2, . . .}

◮ are needed for describing below-zero temperature or debt

The set of rational numbers:

Q =

m

{ n |

m, n ƒ= 0 are integers}

◮ are needed for concepts like half an apple

◮Their corresponding decimals are repeating

The set of irrational numbers: The set of numbers that can not be represented as quo- tients of integers.

◮ are needed to measure distances like the diagonal of a square

◮Their decimal√representations are non- repeating

◮Examples: π, √3.

Question: Why 3 is an irrational number?

The set of real numbers denoted as R is the set of all rational and irrational numbers.

Set Notation and Set Operations

∈: The symbol used to denote that an element is a member of a given set. For example, 2 is an element of the set A = {1, 2, 3}. This is written as 2 ∈ A.

∈/:  The symbol used to denote that an element is not a member of a given set.  For ex- ample, 5 is not an element of the set A. This is written as 5 ∈/A.

If a set A concludes all elements of another set B, it is said that A is a subset of B.    This is written as A ⊂ B.

The union of two sets A and B consists of all elements that are in one or the other of   the sets. The union is denoted as A ∪ B.

The intersection of two sets A and B is the set of all elements that are members of both sets. The intersection is denoted as A ∩ B.

The empty set, denoted by ∅, is the set that contains no element.

Remark:

∅ ⊂ N ⊂ Z ⊂ Q ⊂ R.

Let A B.  The complement of A, denoted by Ac, is the set of all elements in B which are not in A.

Set of Real Numbers

-The set of real numbers denoted as R.

◮ The real numbers are ordered. That is, given a, b R we have either a < b or a > b or

a = b.

◮ We can add, subtract and multiply any two real numbers. We can also divide any number by any non-zero number.

Number Line

The set of real numbers can be represented visually as a line.

  • Each point on the line represents a unique real
  • We choose an arbitrary point to be zero.