SET THEORY: VENN DIAGRAM, REPRESENTATION OF SET AND SOME BASIC OPERATIONS.

Venn diagram

Set can be easily understood and represented by a Venn diagram or set diagram. John Venn used the term `Euler Circles’ to name these diagrams. In order to express the relationships between sets by a simple, instructive and pictorial way is called Euler Venn diagram or shortly Venn diagram. In other word any closed figures used to represent a set is called Venn diagram. It is used to show how things are alike and different. Venn diagram helps student to summarize information, compare things and build comprehension. We represent a set by a simple plane area, usually by a circle.

Example-1: Suppose A⊂B and say, A ≠B. Then A and B can be described by either diagram.

 

Example:2 Suppose A and B are not comparable. Then A and B can be represented by the diagram on the right if they are disjoint, or the diagram on the left if they are not disjoint.

FIGURE-2:Venn diagram when A and B are not comparable

Some basic set operations:

We have learnt to add, subtract and multiply, that is, we assign to each   pair of numbers x and y as x+y which is called the sum of x and y, a number x-y which is called the difference of x and y, and a number xy is called the product of x and y. These assignments are called the operations of addition, subtraction and multiplication of numbers. Now we will learn the operations of union, intersection and difference of sets, that is we will assign new sets to pairs of sets A and B which are somewhat similar to the above operations on numbers.

Set operations: Union

The union of sets A and B is the set of all elements which belongs to A or to B or to both. We denote it by

A U B which is usually read “A union B”. We can show it by the  Venn diagram in figure 1.1 where we have shaded   A U B, i.e. the area of A and the area B.

Example 1: Suppose a set A= {1,2,3,4} and a set B= {1,2,3,4,5,6,7,8}. Here A is a subset of set B

then A U B= {1,2,3,4,5,6,7,8}. We use the common element once time. we can show this set by the a Venn diagram bellow. Here U represent the universal set.

Figure-3:Venn diagram of AUB

 

Figure-4:Set operations:Union

Example 2: Let P is the set of positive real numbers and Q is the numbers of negative real numbers. Then  P U Q, the union of P and Q consists of all real numbers except zero.

The union of A and B may also be defined concisely by

A U B = {x / x ɛ A or x ɛ B}

Note:

1.It follows the union of two sets that   A U B   and   B U A are the same set,i.e.,

A U B= B U A

2.Both A and B are always the subset of   A U B, that is,

A ꞇ (  A U B) and B ꞇ ( A U B)

Set operations: Intersection

The intersection of sets A and B is the set of elements which are common to A and B , that is, those elements which belong to A and which belong s to B. We denote it by

A ∩ B which is read “A intersection B”

Example 1: A= {1,2,3,4,5,6,7,8,9}

 

b= {6,7,8,9,10,11,12,13,14}.here the common elements are 6,7,8 and 9 which is shaded by red color. Then

A ∩ B={6,7,8,9}

Figure-5: A ∩ B={6,7,8,9}

We can show it by the  above Venn diagram:

The intersection of A and B may also be defined concisely by

A ∩ B= { x / x ɛ A and x ɛ B

Note:

  1. It follows directly from the definition of the intersection of two sets that is

A ∩ B= B ∩ A

  1. Each of the sets A and B contains A ∩ B as a subset i.e.,

( A ∩ B) ꞇ A and (A ∩ B) ꞇ B.

  1. If sets A and B have no elements in common, i.e. if A and B are disjoint, then the intersection of A and B is the null set, i.e. A ∩ B= Ø

Set operations: Difference

The difference of sets A and B is the set of elements which belong to A but do not belong to B. We denote the difference of A and B by

A ꟷ  B

Which is read “A difference B” or, simply, “A minus B”

Example 1:

A = {Set of students play cricket} = {a, b, d, e, g, h}

B = {Set of students play football } ={ c, d, f, g, I, k}

A ꟷ B ={Set of students play cricket, not football} ={ a, b, e, h }

B ꟷA = {Set of students play football not cricket} = { c, f, I, k }

In the Venn diagram in the figure-3, we have shaded A ꟷ B,

FIGURE-6: AꟷB is shaded

The difference of A and B may also be defined concisely by

AꟷB =  { x / x ɛ A, x  B }

Note:

Set A contains AꟷB as a subset, i.e.,

AꟷB ꞇ A

The sets (AꟷB),A∩B and (BꟷA) are manually disjoint, that is, the intersection of any two is the null set.

The difference of A and B is sometimes denoted by A/B or A~B.

Set operations: Complement of a set

The complement of a set A is the difference of the universal set U and A, that is, the set of elements which do not belong to A. We denote the complement of A by

A´=UꟷA

Figure-7:Complement of set A ,i.e.  A´

Here we assume that the universal set U consists of the area in the rectangle.

The difference of A and B may also be defined concisely by

A´ =  { x / x ɛ U, x  A }

Or, simply,                A´ =  { x /  x  A }

Note

1 The union of any set A and its complement A´ is the universal set,i.e.,

AUA´=U

Furthermore,

A∩ A´=Ø that is set A and its complement A´ are disjoint set.

2.The complement of the universal set U is the null set Ø, and vice versa.  That is,

U´= Ø and Ø´=U

3.The complement of the complement of a set a A is the set A itself. That is,

(A´)´= A

4.The difference of A and B is equal to the intersection of A and the complement of B,

that is,

AꟷB=A∩B´

It can be easily proved from the definitions:

AꟷB={ x / x ɛ A, x  B } ={ x / x ɛ A, x  B´ } =A∩B´.

 

 

 

 

 

 

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