Relations
`color{blue} ul(mathtt ("Relation"))` `color{blue}:->` A relation `R` from a non-empty set `A` to a non-empty set `B` is a subset of the cartesian product `A × B.` The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in `A × B.`
The second element is called `color{green} ("the image of the first element.")`
`color{purple} ✍️ \color{purple} \mathbf(KEY \ POINTS)`
`color{blue} ul(mathtt ("Domain"))color{blue}:->` The set of all first elements of the ordered pairs in a relation `R` from a set `A` to a set `B` is called `color{green} (" the domain of the relation R.") `
`color{blue} ul(mathtt ("Range")) color{blue}:->` The set of all second elements in a relation `R` from a set `A` to a set `B` is called `color{green} ("the range of the relation R.")`
`color{green} ✍️` The whole set `B` is called `color{green} (" the codomain of the relation R.")`
`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`
● Consider the two sets `P = {a, b, c}` and `Q = {"Ali, Bhanu, Bi noy, Chandra, Divya"}`.
The cartesian product of `P` and `Q` has `15` ordered pairs which can be listed as
`P × Q = {(a, Ali), (a,Bhanu), (a, Bi noy), ..., (c, Divya)}.`
● We can now obtain a subset of `P × Q` by introducing a relation `R` between the first element `x` and the second element `y` of each ordered pair (x, y) as
`R= { (x,y): x \ \ "is the first letter of the name" \ \ y, x ∈ P, y ∈ Q}.`
Then `R = {(a, Ali), (b, Bhanu), (b, Bi noy), (c, "Chandra")}`
A visual representation of this relation `R` (called an arrow diagram) is shown in Fig 1.
`color{fuchsia} "Remarks"`
` (i)` A relation may be represented algebraically either by the Roster method or by the Set-builder method.
`(ii)` An arrow diagram is a visual representation of a relation.
`color{fuchsia} "Note : "` The total number of relations that can be defined from a set `A` to a set `B` is the number of possible subsets of `A × B.` If `color{fuchsia} [n(A ) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia}[n (A × B) = pq]` and `color{fuchsia} "the total number of relations is"` `color{fuchsia} [2^(pq.)]`
The second element is called `color{green} ("the image of the first element.")`
`color{purple} ✍️ \color{purple} \mathbf(KEY \ POINTS)`
`color{blue} ul(mathtt ("Domain"))color{blue}:->` The set of all first elements of the ordered pairs in a relation `R` from a set `A` to a set `B` is called `color{green} (" the domain of the relation R.") `
`color{blue} ul(mathtt ("Range")) color{blue}:->` The set of all second elements in a relation `R` from a set `A` to a set `B` is called `color{green} ("the range of the relation R.")`
`color{green} ✍️` The whole set `B` is called `color{green} (" the codomain of the relation R.")`
`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`
● Consider the two sets `P = {a, b, c}` and `Q = {"Ali, Bhanu, Bi noy, Chandra, Divya"}`.
The cartesian product of `P` and `Q` has `15` ordered pairs which can be listed as
`P × Q = {(a, Ali), (a,Bhanu), (a, Bi noy), ..., (c, Divya)}.`
● We can now obtain a subset of `P × Q` by introducing a relation `R` between the first element `x` and the second element `y` of each ordered pair (x, y) as
`R= { (x,y): x \ \ "is the first letter of the name" \ \ y, x ∈ P, y ∈ Q}.`
Then `R = {(a, Ali), (b, Bhanu), (b, Bi noy), (c, "Chandra")}`
A visual representation of this relation `R` (called an arrow diagram) is shown in Fig 1.
`color{fuchsia} "Remarks"`
` (i)` A relation may be represented algebraically either by the Roster method or by the Set-builder method.
`(ii)` An arrow diagram is a visual representation of a relation.
`color{fuchsia} "Note : "` The total number of relations that can be defined from a set `A` to a set `B` is the number of possible subsets of `A × B.` If `color{fuchsia} [n(A ) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia}[n (A × B) = pq]` and `color{fuchsia} "the total number of relations is"` `color{fuchsia} [2^(pq.)]`
