`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.
`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.