Mathematics Relations

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Relations

Domain and Range of Relation:

If `R` be a relation from a set `A` to set `B`. Then set of all first component's or coordinates of ordered pairs
is called the domain of `R`, while the set of all second component's or coordinates of the ordered pairs is
called as range of relation.

Let `R: A -> B` (`R` is a relation defined from set `A` to set `B`) then domain of this relation is

`text(Domain :)` Set of all the first entries in `R`

`{a | (a, b) in R}`

`text(Range :)` Set of all the second entries in `R`

`{b | (a, b) in R}`

`text(E.g.)` `A = { 1 , 3, 5, 7};` `B = {2, 4, 6, 8}`

Relation is `aRb => a > b;` `a in A, b in B`

`R = {(3, 2), (5, 2), (5 , 3), (7, 2), (7, 4), (7, 6)}`

Domain `= {3, 5, 7}`

Range `= {2 , 4, 6}`

Inverse of a Relation:

lf `R` is a relation defined from `A -> B` then `R^(-1)` is a relation
defined from `B -> A` as

`R^(-1) = {(b, a) | (a, b) inR}`

i.e. domain is converted in to range element's and range is
converted into domain elements.

i.e. Doman of `R =` Range of `R^(-1)`

Range of `R =` Domain of `R^(-1)`

E.g. `N` is a set of first `10` natural nos.

`a R b => a + 2b = 10`

`N = {1 , 2, 3, ... , 10} & a, b in N`

`R = {(2, 4), (4, 3), (6, 2), (8, 1)}`

Inverse relation is `R^(-1) -> { (1 , 8), (2, 6), (3, 4 ), ( 4, 2)}`