# complement of a set | types of set theory | Roster method | universal set

In this article complement of a set we give the information about methods of writing sets and types of set theory like Roster method and Set builder methods. also give information about Equal sets, Subset, Disjoint set, Universal set.

## complement of a set:

The set theory is a branch of mathematics that studies sets. The set theory mainly studies only those related to mathematics.

#### Set Definition :

“A well defined collection of items is called set.”
Examples :
1) My objects : pen, pencil, eraser, ball, book, bag.
2) Even numbers : 2, 4, 6, 8, 10, ……
3) Prime numbers : 2, 3, 5, 7, 11, …….
4) Vowel in English alphabets : a, e, i, o, u.
Individual items in the set is called an elements or a member of the set.
Set are denoted by capital alphabets. such as P, Q, R, X, Y, Z, N, etc.
The elements of the set are generally denoted by small alphabets. such as p, q, r, x, y, z etc.

### Methods of writing sets :

There are two methods of writing sets

1) Listing method / Roster from

2) Rule method / Set builder from

I) Listing method / Roster form :
For example :
The set of letters of the word WRITING
write these distinct alphabets in curly bracket, as { } and separate them by commas, we get the set
D = { w, r, i, t, n, g }
This method of representing a set is known as the Listing method Or Roster form.
Examples :
1) P is the set of numbers whose square is 16.
P = { 4, -4 }
2) Q is the set of first first five multiples of 7.
Q = { 7, 14, 21, 28, 35 }
3) R is the set of first five odd natural numbers.
R = { 1, 3, 5, 7, 9 }
4) S is the set of first 4 prime numbers.
S = { 2, 3, 5, 7 }
II) Rule method / Set builder form :
For example :
consider a set,
A = { 1, 8, 27, 64, 125 }
we observe that there are 5 elements in the set A that are perfect cubes.
The set A can be represented as,
A = { y/y = n × n × n , n = 1 <= n <= 5 }
This is the set builder from of writing set.
Examples :
1) T = { 7, 11, 13, 17, 19, 23 }
Convert into Set builder from
T = { x/x is a prime numbers, 6 < x < 24 }
2) F = { 3, 6, 9, 12 }
Convert into set builder from
E = { y/y is the set of first 4 multiples of 3 }

#### Type of Sets Theory :

There are 4 types of sets
1. Empty set or Null set :
Set having no elements
example:
P = { x/x is even prime numbers greater than two }
P = { }
2. Singleton Set :
Set having only one elements
example :
T = { x/x – 4 = 0 }
i.e. x = 4
3. Finite Set :
Counting of elements terminates at a certain stage.
example :
The set D be the set of 7 first even natural numbers.
D = { 2, 4, 6, 8, 10, 12, 14 }
4. Infinite Set :
Counting of elements do not terminating at any stage.
example :
The set of N of natural numbers.
N = { 1, 2, 3, 4, 5, —–}
Important things :
1) An empty set is a finite Set.
2) N, W, I, Q and R are infinite sets.

#### Equal sets:

“Two sets C and D are said to be equal, if every element of set C is in set D and every element of set D is in set C.”
‘Set C and set D are equal sets’, symbolically it is written as C = D.
Ex (1) C = { x | x is a letter of the word ‘listen’.} Therefor, C = { l, i, s, t, e, n}
D = { y | y is a letter of the word ‘silent’.} Therefor, D = { s, i, l, e, n, t}
Though the elements of set C and D are not in the same order but all the elements are identical. therefor, C = D.
Ex (2) P = {11, 12, 13, 14, 15}
Q = {15, 14, 12, 13, 11}
Though the elements of set P and Q are not in the same order but all the elements are identical. therefor, P = Q.
This can be represented by Venn diagram as shown below:

#### Subset:

If C and D are two given sets and every element of set D is also an element of set C then
D is a subset of C which is symbolically written as D ⊆ C. It is read as ‘D is a subset of C’ or ‘D subset C’.
Ex 1.
C = { 11, 12, 13, 14, 15, 16, 17, 18}
D = {12, 14, 16, 18}
Every element of set D is also an element of set C.
Therefor, D ⊆ C.
This can be represented by Venn diagram as shown below:

#### Universal set:

• The Universal set means all elements are inside in the set that is a whole set which will accommodate all the given sets under consideration which in general is known as Universal set.
• So that the sets under consideration are the all subsets of this Universal set.
• Universal set are indicated by using the U latter and use a rectangle box.

Example: Suppose we want to study the students in class FY BCA who frequently remained absent. Then we have to think of all the students of class FY BCA who are in the College. So all the students in a school or the students of all the divisions of class FY BCA in the College is the Universal set.

Disjoint sets:
Let, P = { 11, 13, 15, 19}
and Q = {22, 24, 28} are given.
Confirm that not a single element is common in set P and Q. These sets are completely different from each other. So the set P and Q are disjoint set.

This can be represented by Venn diagram as shown below: –

#### Complement of a set:

Suppose U is an universal set. If P⊆U, then the set of all elements in U, which are not in set P is called the complement of P. It is denoted by  P′.
P′ is defined as follows.
P′ = {x | x ⊂ U, and x ⊄ B}
Ex (1) U = { 41, 42, 43, 44, 45, 46, 47, 48, 49, 50}
P = {42, 44, 46, 48, 50}
P′ = {41, 43, 45, 47, 49}

This can be represented by Venn diagram as shown below: – 