`color {green}{★ ul"Cartesian Products of Sets"}`

Given two non-empty sets `P` and `Q.` The cartesian product `P × Q` is the set of all ordered pairs of elements from `P` and `Q,` i.e.,

` \ \ \ \ \ \ color{green} (P × Q = { (p,q) : p ∈ P, q ∈ Q })`

If either `P` or `Q` is the null set, then `P × Q` will also be empty set, i.e., `P × Q = φ`

In most of the cases `P xx Q ne Q xx P`

`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{green} ✍️` Note that `color{green} "range ⊆ codomain."`

`\color{blue} ☛\color{blue} ul("Function") `

A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.

`\color{blue} ☛ color{blue} ul ("Real valued function")`

A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.

`\color{blue} ☛ color{blue} ul ("Identity function")`

Define the real valued function `f : R → R` by `y = f(x) = x` for each `x ∈ R.` Such a function is called the identity function.

`\color{blue} ☛ color{blue} ul ("Constant function")`

Define the function `f: R → R` by `y = f (x) = c, x ∈ R` where `c` is a constant and each `x ∈ R.`

`\color{blue} ☛ color{blue} ul ("Polynomial function")`

`y = f (x) = a_0 + a_1x + a_2x^2 + ...+ a_n x^n,` where `n` is a non-negative integer and `a_0, a_1, a_2,...,a_n ∈R.`

`\color{blue} ☛ color{blue} ul ("Rational functions")`

Rational functions are functions of the type `(f( x))/ (g (x))` ,

where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0.`

`\color{blue} ☛ color{blue} ul ("The Modulus function")`

The function `f: R→R` defined by `f(x) = |x|` for each `x ∈R` is called modulus function.

For each non-negative value of `x, f(x)` is equal to `x.` But for negative values of `x`, the value of `f(x)` is the negative of the value of `x,` i.e.,

`f(x)= { tt (( x , if , x >= 0),(-x, if, x< 0))`

`\color{blue} ☛ color{blue} ul ("Signum function ")`

Signum function The function `f:R→R` defined by

`f(x)= { tt (( 1, if , x > 0),(0, if, x=0),( -1, if , x < 0))`

is called the signum function.

`\color{blue} ☛ color{blue} ul ("Greatest integer function")`

`f(x) = [x], x ∈R` assumes the value of the greatest integer, less than or equal to `x.` Such a function is called the greatest integer function.

From the definition of `[x],` we can see that

`tt (( [x]= -1, "for" \ \ –1 ≤ x < 0 ),([x]=0 , "for" \ \ 0 ≤ x < 1 ),( [x] = 1 , "for" \ \ 1 ≤ x < 2 ),([x] = 2, "for" \ \ 2 ≤ x < 3))`

and so on.

Given two non-empty sets `P` and `Q.` The cartesian product `P × Q` is the set of all ordered pairs of elements from `P` and `Q,` i.e.,

` \ \ \ \ \ \ color{green} (P × Q = { (p,q) : p ∈ P, q ∈ Q })`

If either `P` or `Q` is the null set, then `P × Q` will also be empty set, i.e., `P × Q = φ`

In most of the cases `P xx Q ne Q xx P`

`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{green} ✍️` Note that `color{green} "range ⊆ codomain."`

`\color{blue} ☛\color{blue} ul("Function") `

A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.

`\color{blue} ☛ color{blue} ul ("Real valued function")`

A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.

`\color{blue} ☛ color{blue} ul ("Identity function")`

Define the real valued function `f : R → R` by `y = f(x) = x` for each `x ∈ R.` Such a function is called the identity function.

`\color{blue} ☛ color{blue} ul ("Constant function")`

Define the function `f: R → R` by `y = f (x) = c, x ∈ R` where `c` is a constant and each `x ∈ R.`

`\color{blue} ☛ color{blue} ul ("Polynomial function")`

`y = f (x) = a_0 + a_1x + a_2x^2 + ...+ a_n x^n,` where `n` is a non-negative integer and `a_0, a_1, a_2,...,a_n ∈R.`

`\color{blue} ☛ color{blue} ul ("Rational functions")`

Rational functions are functions of the type `(f( x))/ (g (x))` ,

where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0.`

`\color{blue} ☛ color{blue} ul ("The Modulus function")`

The function `f: R→R` defined by `f(x) = |x|` for each `x ∈R` is called modulus function.

For each non-negative value of `x, f(x)` is equal to `x.` But for negative values of `x`, the value of `f(x)` is the negative of the value of `x,` i.e.,

`f(x)= { tt (( x , if , x >= 0),(-x, if, x< 0))`

`\color{blue} ☛ color{blue} ul ("Signum function ")`

Signum function The function `f:R→R` defined by

`f(x)= { tt (( 1, if , x > 0),(0, if, x=0),( -1, if , x < 0))`

is called the signum function.

`\color{blue} ☛ color{blue} ul ("Greatest integer function")`

`f(x) = [x], x ∈R` assumes the value of the greatest integer, less than or equal to `x.` Such a function is called the greatest integer function.

From the definition of `[x],` we can see that

`tt (( [x]= -1, "for" \ \ –1 ≤ x < 0 ),([x]=0 , "for" \ \ 0 ≤ x < 1 ),( [x] = 1 , "for" \ \ 1 ≤ x < 2 ),([x] = 2, "for" \ \ 2 ≤ x < 3))`

and so on.

`color{green} ✍️` Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.

`color{green} ✍️` If `A` and `B` are non-empty sets and either `A` or `B` is an infinite set, then so is `A × B.`

`color{green} ✍️ \ \ A × A × A = {(a, b, c) : a, b, c ∈ A}.` Here `(a, b, c)` is called an ordered triplet.

`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} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`

`color{green} ✍️` If `A` and `B` are non-empty sets and either `A` or `B` is an infinite set, then so is `A × B.`

`color{green} ✍️ \ \ A × A × A = {(a, b, c) : a, b, c ∈ A}.` Here `(a, b, c)` is called an ordered triplet.

`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} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`

`color{green} ✍️` In most of the cases `P xx Q ne Q xx P`

`color{green} ✍️` If there are `p` elements in `A` and `q` elements in `B`, then there will be `pq` elements in `A × B,` i.e., if `color{fuchsia} [n(A) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia} [n(A × B) = pq.]`

`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`

`color{green} ✍️` 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.)]`

`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`

`color{green} ✍️` If there are `p` elements in `A` and `q` elements in `B`, then there will be `pq` elements in `A × B,` i.e., if `color{fuchsia} [n(A) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia} [n(A × B) = pq.]`

`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`

`color{green} ✍️` 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.)]`

`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`