# Types of sets: Null set, Singleton set, Finite and infinite set, Subsets, Universal set and Power set.

**WEEK FOUR**

**SS1 FURTHER MATHS FIRST TERM**

**SETS**

CONTENTS:

- Definition of set
- Set notation methods
- Types of sets: Null set, Singleton set, Finite and infinite set, Subsets, Universal set and Power set.
- Law of algebra of sets

**SUB TOPIC: DEFINITION OF SET**

A set is a collection of distinct objects, things, objects or numbers.

Examples includes:

- A collection of books in public or private library
- A collection of tools in a carpentry workshop
- Letters of the alphabet
- The collection of all positive numbers

Basically we can conclude that any clearly defined collection of things, objects or numbers constitutes a set.

When the number of elements in a given set is countable, the set is called a FINITE SET. E.g whole numbers lying between 1 and 10, members of a football team, etc. but if the number of elements in a set is not finite/countable, then the set is called an INFINITE SET. E.g real numbers, rational numbers, etc.

**CLASS ACTIVITIES:**

- Define set
- List five (5) examples of sets
- Differentiate between a finite set and an infinite set with five (5)examples each.

**SUB TOPIC: SET NOTATION METHODS**

Each member of a set is called an element of the set. Capital letters will be used to denote sets while small letters will be used to denote the elements of a set.

When an element belongs to a set , we write and say that is a member or element of If is not a member or element of we write and say that does not belong to .

**Example: **

If is the set of all prime numbers and then . If , then .

A set is completely specified in the following ways:

- By lising all the members of the set;
- By describing the elements of the sets;
- By enclosing the members or elements in curly bracket or braces: {} or .

**CLASS ACTIVITIES:**

- (a). What is an element of a set? (b). Write out another name for element of a set
- What is the meaning of the following signs in set notations:
- . (b).
- If is the set of even numbers and , then is what to

**SUB TOPIC: TYPES OF SET**

**Null or empty set:**When a set contains no element or member is said to be null or empty set and denoted by .

**Examples:**

- Human beings with tails}
- {all prime members of SS1 class divisible by 4}
**Singleton or unit set:**A set which consist of only one element or member is called unit or singleton set.

**Example:** is a unit or singleton set because it contains only one element.

**Finite set:**When the number of elements in a given set is finite can be be listed or stated.

**Example: **

is finite.

The members of the set

**Infinite set:**If the number of element in a set is not finite or cannot be listed.

**Example: **

, since we can not list all the element in the set

**Subset:**if all the elements of the set are members of a set the set is a subset of We denote this by a set notation. Thus, means is a subset of**A proper subset of**If the set is a subset of and there is at least one element in the set which is not in then the set is called a proper subset of or is superset of set . Thus means is a proper subset of Thus means is a superset of**Universal set:**The set which contains all the possible elements under consideration. The universal set is denoted by .**Power set:**The set of all subsets of a set The power set is usually denoted by

**CLASS ACTIVITIES:**

- Mention the types of sets you know.
- Explain with example a universal set.
- What is a subset?
- Explain the difference between a proper subset and superset.

**SUB TOPIC: LAWS OF ALGEBRA OF SET**

Consider the diagram below involving three sets

**Commutative Laws**

**Associative Laws**

**Distributive Laws**

**Idempotent Laws**

**De Morgan’s Laws**

**Laws of Complementation **

**Additional Properties**

**Laws of Inclusion**

PRACTICE EXERCISE:

**Objective Test:**

- ………………is a collection of distinct objects, things or numbers
- Set B. elements C. members D. universal E. notation
- Each member of a set is called an……………..
- Subset B. superset C. elements D. finite E. singleton
- Which bracket is use in enclosing the elements of a set?
- Big bracket B. small bracket C. open bracket D. curly bracket E. set bracket.
- The sign of subset is………… A. B. C. D. E.
- …………….set is a which contains all the possible elements under consideration?
- Universal B. power C. finite D. singleton E. null

**Essay Questions**

{all the days in a week}; {all the days whose letters begin with T}

- List all the elements of
- List the members of
- List the members of
- What is ?
- What is ?

**KEY WORDS**

**SETS****COLLECTION****ELEMENT****MEMBER****UNION****INTERSECTION****UNIVERSAL****PRIME/COMPLEMENT****VENN DIAGRAM****SUBSET****NULL SET****POWER SET****CARDINAL SET****NOTATION****OPERATIONS**