Cartesian Product of Sets
* Ordered pair: A pair entries grouped in a particular order, which are separated by a comma and enclosed within brackets.
* If two ordered pairs are equal then their corresponding first elements and second elements are equal.
* The Cartesian product of two non-empty finite sets P and Q is the set of all ordered pairs of the elements from P and Q.
* In set-builder form,
P × Q = {(p, q): p ∈ P, q ∈ Q}
* The Cartesian product of two non-empty sets X and Y can be represented in the form of tabular cells or the point of intersection of perpendicular lines.
* If n(P) = m and n(Q) = then n(P × Q) = mn.
Relations
* A relation gives a connection or an association between objects, ideas, quantities or individuals.
* A relation can be described in different ways:
1. List form or Roster form
2. Set-builder form
3. Arrow diagram (visual representation)
* A relation from a non-empty set A to a non-empty set B can be defined as the subset A × B, which is obtained by describing a relationship between the first
element and the second element of the ordered pairs.
* The set of the first coordinates of all the ordered pairs of a relation R is called the domain of R.
* The set of the second coordinates of all the ordered pairs of R is called the range of R.
* The total numbers of relations that can be defined from a set A to a set B is the number of possible subsets of A × B.
* If n(A) = p and n(B) = q, then n(A × B) = pq and the total number of relations is 2pq.
Functions
* A function that has either R, (the set of real numbers) the set of real numbers, or one of its subsets as its range is called a Real Valued Function.
* If the domain and the range of the function is either R or a subset of R, then it is called a Real function.
Functions and their Graphs
* A function ƒ: A → A is said to be an Identity function if f(x) = X, ∀ X ∈ A.
Let ƒ: R → R be a function defined by f(x) = ax + b where a ≠ 0, a and b are real constants and x is a real variable. Such a function is known as the Linear function.
* A function ƒ: R → R is said to be a constant function, if ƒ(x) = c, ∀ x∈ R, where c is a constant.
* A function ƒ: R → R is said to be polynomial function is for all x in R, y = ƒ(x) =a0 + a1x +a2x2+ … + anxn, where n is a non negative integer and a0, a1, a2, … an are real numbers.
* Let ƒ and g be two polynomial functions, then \frac{f(x)}{g(x)} where ɡ(x) ≠ 0 and x ∈ R, is known as Rational function.
* A function ƒ: R → R defined by f(x) = |x | ∀x ∈ R is known as modulus function.
* A function ƒ: R → R defined by f(x) = \frac{|x|}{x} is called a Signum function.
* The function ƒ: R → R defined by f(x) = [x], x ∈ R assumes the value of the greatest integer function.