Relations and Sets Formulas

Collection of objects which is well-defined is known as a Set. Generally set is synonym of many words which are used in mathematics as: collection, aggregate, class and all these are comprised of elements. If two Sets are related to each other by Subsets then it is known as relation. There are different Relations and Sets Formulas which are shown below:
For set, X U X = X, X ∩ X = X and X ∩ U = X for identity,
X U Y = Y U X, X ∩ Y = Y ∩ X for commutative,
(X ∩ Y) ∩ Z = X ∩ (Y ∩ Z) for associative.
De-Morgan’s law: (X U Y)’ = X’ ∩ Y’,
these are the Basic Formulas.
n (X U Y) = [n(X) + n(Y) – n(X ∩ Y)] and
n(X ∩ Y)’ = n(U) – n(X U Y),
For relations:
Formulas of Domain and range of relation: Dom (R) = X: (X, Y) ϵ R and Range(R) = Y: (X, Y) ϵ R, where Dom indicates the domain of relations. In this definition domain of a relation from X to Y is a subset of 'X' and range of relation is subset of 'Y'.
Reflexivity formula: (X, Y) ϵ N * N = X, Y = N and X + Y = Y + X.
Some Sets and Relations problems are shown below:
If X, Y are any sets and 'U' is a universal set, where n (U) = 500, n(X) = 100, n(Y) = 200 and n(X ∩ Y) = 50. Find n(X’∩Y’)?
We have, X’∩ Y’ = (X ∩ Y)’
 n(X’∩Y’) = n(X∩Y)’ = n(U) – n(X ᴜ Y) = n(U) – [n(X) + n(Y) – n(X∩Y)],
n(X’∩Y’) = 500 – [100+200-50] = 500 – 250 = 250.
If X = 1, 3, 5 and Y = 4, 6, 8, 10 and R = (1, 8), (3, 6) (1, 10) is the relation from X to Y, then find the domain and range.
Dom(R) = 1, 3 and The Range(R) = 8, 6, 10.