Union and Intersection of sets
`(A)text(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 `cup` is used to denote the union. Symbolically, we write `A cup B` and usually read as `-A union B-.` For Example Let `A = { 2, 4, 6, 8} ` and `B = { 6, 8, 10, 12}.` We have `Acup B = { 2, 4, 6, 8, 10, 12}` `text(Note that)` the common elements `6` and `8` have been taken only once while writing `A ∪ B.`
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 in B }` The union of two sets can be represented by a Venn diagram as shown in `Fig (a).` The shaded portion in `Fig(a)` represents `A ∪ B.`
`text(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 ∪ phi = A` (Law of identity element, `phi` is the identity of `∪)`
`(iv)`` A ∪ A = A` (Idempotent law)
`(v)`` U ∪ A = U (`Law of `U)`
`(B)text(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}.`
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 (b)` indicates the interseciton of `A` and `B.`
`text(Some Properties of Operation of Intersection)`
`(i) ``A ∩ B = B ∩ A ` (Commutative law).
`(ii)`` ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )` (Associative law).
`(iii)`` phi ∩ A = phi , \ \ \ \ \ \ U ∩ A = A (`Law of `phi` and `U).`
`(iv)`` A ∩ A = A(`Idempotent law`)`
`(v)` `A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) `(Distributive law ) i. e., `∩` distributes over `∪`
`(C)text(Disjoint sets.)`
If `A` and `B` are two sets such that `A ∩ B = phi`, then `A` and `B` are called `text(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 (c)` In the above diagram, `A` and `B` are disjoint 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 `cup` is used to denote the union. Symbolically, we write `A cup B` and usually read as `-A union B-.` For Example Let `A = { 2, 4, 6, 8} ` and `B = { 6, 8, 10, 12}.` We have `Acup B = { 2, 4, 6, 8, 10, 12}` `text(Note that)` the common elements `6` and `8` have been taken only once while writing `A ∪ B.`
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 in B }` The union of two sets can be represented by a Venn diagram as shown in `Fig (a).` The shaded portion in `Fig(a)` represents `A ∪ B.`
`text(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 ∪ phi = A` (Law of identity element, `phi` is the identity of `∪)`
`(iv)`` A ∪ A = A` (Idempotent law)
`(v)`` U ∪ A = U (`Law of `U)`
`(B)text(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}.`
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 (b)` indicates the interseciton of `A` and `B.`
`text(Some Properties of Operation of Intersection)`
`(i) ``A ∩ B = B ∩ A ` (Commutative law).
`(ii)`` ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )` (Associative law).
`(iii)`` phi ∩ A = phi , \ \ \ \ \ \ U ∩ A = A (`Law of `phi` and `U).`
`(iv)`` A ∩ A = A(`Idempotent law`)`
`(v)` `A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) `(Distributive law ) i. e., `∩` distributes over `∪`
`(C)text(Disjoint sets.)`
If `A` and `B` are two sets such that `A ∩ B = phi`, then `A` and `B` are called `text(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 (c)` In the above diagram, `A` and `B` are disjoint sets.