`\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.`

`\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`)

`\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.

`=>` 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`

(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)].

`\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"`