NOTION OF A

SET

SET

A SET is a well-defined collection of objects, real or imagined.

NOTION OF A SET

Examples:

The collection of all even counting numbers.

The collection of casts in Pure Love.

REMEMBER:

A set is well-defined if it is possible to determine whether a given object belongs to the set or not.

TIME TO THINK!

Which of the following is a set?

1. Collection of all barangays in Los Baos

2. Collection of all numbers greater than 100

3. Collection of all tall trees in the Makiling Botanic Garden

4. All large positive numbers

Given the set of all provinces in the Philippines, which are included in the set?

Ilo-ilo?

Bohol?

Sorsogon?

Saranggani?

Batangas City?

Buenos Aires?

To name a set, we usually use capital letters.

For the given set, the set of all provinces in the Philippines, we will use P.

If an object belongs to the set, it is called an element of the set.

In symbols, we write , if a is an element of the set .

Otherwise, we write if a is NOT an element of the set A.

a AA

a A

Thus, in our example, we write

whileLaguna P NewYork P

ROSTER METHOD. Elements of the set are listed or enumerated then enclosed within braces.

Describing Sets

RULE METHOD. Enclosed within braces is a phrase describing the elements of the set with the condition that those object and only those which have the described property belong to the set.

Describing Sets

under RULE METHOD. The set-builder notation is a way of writing sets using rule method. In set builder notation we write read as set of all x such that x is a ________.

| is a __________x x

Example. Consider C= the collection of all even numbers between 1 and 10.

C = { 2,4,6,8}

Describing Sets

By the Rule Method, C can also be described as

By Roster Method, C can be described as

C | is an even number between 1 and 10x x

Example. The set of all RGEP courses offered in UPLB, denoted by G, can be written using the rule method in the following way:

Name some elements of G.

in UPLB offered course RGEPan is| xxG

Describe the following sets using the roster method:

Set of all vowels found in the English alphabet

Set of all recitation teachers of MATH 11 D of 1st semester AY 2014-2015

TIME TO THINK!

When is the rule method more advantageous to use than the roster method?

When is the roster method more useful than the rule method?

Special Sets

A set without any element is called an empty (or null) set. It is denoted by { } or .

The set of all elements under consideration is called the universal set. It is denoted by U.

Examples of empty set are:

1. The set of snakes (in the real world) that have hands.

2. The set of all negative numbers greater than zero

For example, if we consider M={red, orange, blue},

then a possible universal set for M is the set of all colors of the rainbow.

Thus U = set of all colors of the rainbow.

For example, if we consider S={1,2,3,4},

then a possible universal set for S is the set of all counting numbers.

Thus U = set of all counting numbers.

Another possible universal set for S is the set of all positive numbers.

Thus U = set of all postive numbers.

Remark:

There can be many possible universal sets for a given set.

NOTION OF A SET

Is this an empty set?

Identify some more empty sets.

TIME TO THINK!

{}

Give a universal set from which the members of the following sets could be chosen:

1. { 3, 5, 7, 9}

2. {BS DevCom, BS Agri, BS Forestry, BA ComArts}

3. { ABS-CBN, GMA}

TIME TO THINK!

In this section, we learned the following:

When a set is well-defined or not

Sets are denoted by capital letters

Objects belonging to a set are called elements

If a belongs to set B we write a B.

Otherwise, we write a B.

Sets can either be described by the rule or the roster methods

There are two special sets, namely, empty sets and universal sets.

For recitation class on Tuesday

Notion of a Set