Venn Diagrams and Operation on Sets (Set theory Part 7 NCERT)

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Venn Diagrams

Most of the relationships between sets can be represented by means of diagrams which are known asVenn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles. 
In Venn diagrams, the elements of the sets are written in their respective circles (Figs 1.2 and 1.3) 

Illustration 1 In Fig 1.2, U = {1,2,3, …, 10} is the universal set of which A = {2,4,6,8,10} is a subset. 
1.2 Illustration 2 In Fig 1.3, U = {1,2,3, …, 10} is the universal set of which A = {2,4,6,8,10} and B = {4, 6} are subsets, and also B ⊂ A. 

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Operations on Sets

In earlier classes, we have learnt how to perform the operations of addition, subtraction, multiplication and division on numbers. Each one of these operations was performed on a pair of numbers to get another number. For example, when we perform the operation of addition on the pair of numbers 5 and 13, we get the number 18. Again, performing the operation of multiplication on the pair of numbers 5 and 13, we get 65. Similarly, there are some operations which when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer all our sets as subsets of some universal set.

Union of sets 
Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘∪’ is used to denote the union. Symbolically, we write A ∪ B and usually read as ‘A union B’. 

Example 12 Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B. 
Solution We have A ∪ B = { 2, 4, 6, 8, 10, 12} 
Note that the common elements 6 and 8 have been taken only once while writing A ∪ B. 

Example 13 Let A = { a, e, i, o, u } and B = { a, i, u }. Show that A ∪ B = A 
Solution We have, A ∪ B = { a, e, i, o, u } = A. 
This example illustrates that union of sets A and its subset B is the set A itself, i.e., if B ⊂ A, then A ∪ B = A. 

Example 14 Let X = {Ram, Geeta, Akbar} be the set of students of Class XI, who are in school hockey team. Let Y = {Geeta, David, Ashok} be the set of students from Class XI who are in the school football team. Find X ∪Y and interpret the set. 
Solution We have, X ∪ Y = {Ram, Geeta, Akbar, David, Ashok}. This is the set of students from Class XI who are in the hockey team or the football team or both. 

Thus, we can define the union of two sets as follows: 
Definition 6 The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write. 
A ∪ B = { x : x ∈ A or x ∈ B } 
The union of two sets can be represented by a Venn diagram as shown in Fig 1.4.The shaded portion in Fig 1.4 represents A ∪ B. 
 

1.4
Some Properties of the Operation of Union 
(i) A ∪ B = B ∪ A (Commutative law) 
(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law ) 
(iii) A ∪ φ= A (Law of identity element, φ is the identity of ∪) 
(iv) A ∪ A = A (Idempotent law) 
(v) U ∪ A = U (Law of U)

Intersection of sets 
The intersection of sets A and B is the set of all elements which are common to both A and B. The symbol ‘∩’ is used to denote the intersection. The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}. 

Example 15 Consider the sets A and B of Example 12. Find A ∩ B. 
Solution We see that 6, 8 are the only elements which are common to both A and B. Hence A ∩ B = { 6, 8 }. 

Example 16 Consider the sets X and Y of Example 14. Find X ∩ Y. 
Solution We see that element ‘Geeta’ is the only element common to both. Hence, X ∩ Y = {Geeta}. 

Example 17 Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = { 2, 3, 5, 7 }. Find A ∩ B and hence show that A ∩ B = B. 
Solution We have A ∩ B = { 2, 3, 5, 7 } = B. We note that B ⊂ A and that A ∩ B = B. 

Definition 7 The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B} The shaded portion in Fig 1.5 indicates the interseciton of A and B. 

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If A and B are two sets such that A ∩ B = φ, then A and B are called disjoint sets. For example, let A = { 2, 4, 6, 8 } and B = { 1, 3, 5, 7 }. Then A and B are disjoint sets, because there are no elements which are common to A and B. The disjoint sets can be represented by means of Venn diagram as shown in the Fig 1.6 In the above diagram, A and B are disjoint sets. 
1.6 

Some Properties of Operation of Intersection 
(i) A ∩ B = B ∩ A (Commutative law) 
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C) (Associative law ) 
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U). 
(iv) A ∩ A = A (Idempotent law) 
(v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e.,∩ distributes over ∪ 

This can be seen easily from the following Venn diagrams [Figs 1.7 (i) to (v)]. 

1.7

 

Difference of sets 
The difference of the sets A and B in this order is the set of elements which belong to A but not to B. Symbolically, we write A – B and read as “ A minus B”. 
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Example 18 Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A. 
Solution We have, A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B and B – A = { 8 }, since the element 8 belongs to B and not to A. We note that A – B ≠ B – A. 

Example 19 Let V = { a, e, i, o, u } and B = { a, i, k, u}. Find V – B and B – V 
Solution We have, V – B = { e, o }, since the elements e, o belong to V but not to B and B – V = { k }, since the element k belongs to B but not to V. We note that V – B ≠ B – V. Using the setbuilder notation, we can rewrite the definition of difference as 
A – B = { x : x ∈ A and x ∉ B } The difference of two sets A and B can be represented by Venn diagram as shown in Fig 1.8. The shaded portion represents the difference of the two sets A and B. 

Remark The sets A – B, A ∩ B and B – A are mutually disjoint sets, i.e., the intersection of any of these two sets is the null set as shown in Fig 1.9. 

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