Mathematics Definitions , Key Concepts, Points to Remember & Formulas

### Definitions

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.

### Key Concepts

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.

### Points to Remember

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.