Points to Remember: Relation and Function

Key Concepts

The Cartesian Product of \(A\) with \(B\) is defined as \(A \times B = \{(a, b) \mid a \in A, b \in B\}\).
A relation R from \(A\) to \(B\) is always a subset of \(A \times B\). That is, \(R \subseteq A \times B\).
A relation R from \(X\) to \(Y\) is a function if for every \(x \in X\) there exists only one \(y \in Y\).
A function can be represented by:
- (i) an arrow diagram
- (ii) a tabular form
- (iii) a set of ordered pairs
- (iv) a graphical form
Some types of functions:
- (i) One-one function
- (ii) Onto function
- (iii) Many-one function
- (iv) Into function
Identity function: \(f(x) = x\)
Reciprocal function: \(f(x) = \frac{1}{x}\)
Constant function: \(f(x) = c\)
Linear function: \(f(x) = ax + b, a \neq 0\)
Quadratic function: \(f(x) = ax^2 + bx + c, a \neq 0\)
Cubic function: \(f(x) = ax^3 + bx^2 + cx + d, a \neq 0\)
For three non-empty sets \(A\), \(B\) and \(C\), if \(f : A \to B\) and \(g : B \to C\) are two functions, then the composition of \(f\) and \(g\) is a function \(g \circ f : A \to C\) will be defined as \((g \circ f)(x) = g(f(x))\) for all \(x \in A\).
If \(f\) and \(g\) are any two functions, then in general, \(f \circ g \neq g \circ f\).
If \(f\), \(g\) and \(h\) are any three functions, then \(f \circ (g \circ h) = (f \circ g) \circ h\).