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.