Mathematics Venn Diagrams, Union of sets, Intersection of sets and Difference of sets

### Topics Covered

star Venn Diagrams
star Union of sets
star Intersection of sets
star Difference of sets

### Venn Diagrams :

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"

• A \color{blue} ul(\mathtt ( \ \ VEN N \ \ Diagram)) is a diagram that uses closed curves to illustrate the relationships among sets.

• These diagrams consist of rectangles and closed curves usually circles or ellipses. The universal set is represented usually by a rectangle and its subsets by circles or ellipses.

E.g. 1 : In Fiig.1 \ \  U = {1,2,3, ..., 10} is the universal set of which A = {2,4,6,8,10} is a subset.

E.g. 2 In Fig ,2\ \  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.

### Union of sets :

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"

• Let A and B be any two sets. \color{blue} ul(\mathtt (THE \ \ UNION \ \ OF\ \ A\ \ AND \ \ B)) is the set which consists of \color{blue} ul(\mathtt ( "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’."

"Definition" 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).

•The common elements being taken only once.
A ∪ B = { x : x ∈ A or x ∈B }

The union of two sets can be represented by a Venn diagram as shown in Fig 3 . The shaded portion in Fig 3 represents A ∪ B.

Eg., Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.

A ∪ B = { 2, 4, 6, 8, 10, 12}

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)

Q 3057767684

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.
Q 3087767687

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.
Q 3027867781

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

### Intersection of sets :

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"

• The intersection of two sets A and B is the set of all those elements which \color{blue} ul(\mathtt ("belong to both A and B.")

• The symbol ‘∩’ is used to denote the intersection.

• Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}

The shaded portion in Fig 4. indicates the interseciton of A and B.

star \ \ "Disjoint sets"

• If A and B are two sets such that A ∩ B = Phi , 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 5.

Q 3007867788

Consider the sets A and B of A={ 2, 4, 6, 8} and B = { 6, 8, 10, 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 }.
Q 3077067886

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.

Solution:

We see that element ‘Geeta’ is the only element common to both. Hence, X ∩ Y = {Geeta}.
Q 3017067889

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.

### Properties Of Operations

=> Commutative Law :

If a star b = b star a => "operation" star "follows commutative Law".

=> Associative Law :

If (a star b) star c= (a star b) star c, then star" follows associative Law".

=> Distributive Law:

7 xx (4+3)=(7 xx4 )+(7 xx 3)

If a star (b)= a star b + a star c  , then star" follows Distributive law".

Eg., A cup (B cap C)=(A cup B ) cap (A cup C)

=> Idempotent Law:

If a star a= a forall a , then star" follows Idenpotent Law".

=> Identity Element :

If a star b= a forall a, then b is called identity elements.

0 is identity element for addition 5+0=5
1 is identity element for multiplication 5 xx 1=5

### Some Properties of Operation of Union 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 ∪(Proof is shown in Fig. )

(v) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

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

### Difference of sets (A-B) or (A/B) :

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"

• The difference of the sets A and B in this order is the set of elements which \color{blue} ul(\mathtt ( "belong to A but not to B.")

• Symbolically, we write A – B and read as "“ A minus B”"

Eg. Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }.

A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B
B – A = { 8 }, since the element 8 belongs to B and not to A.

 \color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION ) :  We note that A – B ≠ B – A.

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)

• 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.

\color { maroon} ® \color{maroon} ul (REMEMBER) " A-B also represented as A/B"
Q 3067167985

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.
Q 3027178081

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 set builder 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.