DHARMDISCRETE THEORY, RELATIONS AND FUNCTIONS 5i.e.
X ∪ Y = {x/x ∈ X OR x ∈ Y}
For example,
l {a,, b, c} ∪ {1, 2} = {a, b, c, 1, 2} and others.
l {a,, b, c} ∪ {a, b, c} = {a, b, c}
l {a,, b, c} ∪ { } or ∅ = {a, b, c}
l {a,, b, c} ∪ {{a, b}, c} = {a, b, {a, b}} or {{a, b}} and others.
In general we conclude that elements of the union of the sets have at least one of the
property embraces by elements of set X or set Y.
Similar to the union of two sets, if there are n-sets X 1 , X 2 , ......... Xn then there combina-
tion using union operator is denoted by,
X 1 ∪ X 2 ∪ ........Xn = (....((X 1 ∪ X 2 ) ∪ X 3 ) ........Xn) = kn=to1∪ Xk
where, set Xk is also called indexed-set.
Intersection
Given two sets X and Y, then intersection of X and Y denoted by X ∩ Y is the set that has all
the common elements of both the sets X and Y, i.e.,
X ∩ Y = {x/x ∈ X and x ∈ Y}
For example,
l {a, b, c} ∩ {a, b, c, d, e} = {a, b, c}
Two sets are said to be disjoint if they have no common element, i.e.,
l {a, b, c} ∩ {1, 2} = { } or ∅
l {a, b, c} ∩ { } = { } or ∅
l {∅, a, b, c} ∩ { } = { } or ∅
l But {∅, a, b, c} ∩ {{ }} = {∅} is not disjoint sets.
In general, the elements of the intersection of the sets embrace both the elements prop-
erty of the set X as well as the elements property of the set Y.
In the similar mode we can combine the n-sets (X 1 , X 2 , ......... Xn) using intersection
operations as,
X 1 ∩ X 2 ∩ ........Xn = (....((X 1 ∩ X 2 ) ∩ X 3 ) ........Xn) = ∪
kn=to1
Xk
where, set Xk is an indexed-set.
We can also see that union and intersection operations are commutative and associative,
i.e.
(i) X ∪ Y = Y ∪ X and (ii) X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
also (i) X ∩ Y = Y ∩ X and (ii) X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z
Example 1.1. Set X = {x/x is an integer s.t. x ≥ 10}, Y = {1, 2, 3, .....} and Z = {3, 5, 7, 9} find
X ∪ Y, X ∪ Z, X ∩ Y and X ∩ Z.
Sol. X ∪ Y = {1, 2, 3,.....}
[This set contains all elements of set X and all elements of set Y].
X ∪ Z = {3, 5, 7, 9, 10, 11, 12, .....}.
X ∩ Y = {10, 11,......} and X ∩ Z = ∅.