Class 11: Relations and Functions

Cartesian Product, Relations, Functions & Algebra of Functions

Ordered Pair	A pair of elements grouped together in a particular order. Two ordered pairs $$(x, y)$$ and $$(a, b)$$ are equal if and only if $$x = a$$ and $$y = b$$.
Definition	Given two non-empty sets $$P$$ and $$Q$$. The cartesian product $$P \times Q$$ is the set of all ordered pairs of elements from $$P$$ and $$Q$$. $$P \times Q = \{(p,q) : p \in P, q \in Q\}$$.
Number of Elements	If $$n(A) = p$$ and $$n(B) = q$$, then $$n(A \times B) = pq$$.
Properties	1. If $$A$$ or $$B$$ is null, then $$A \times B = \phi$$. 2. Generally, $$A \times B \neq B \times A$$. 3. $$A \times A \times A = \{(a, b, c) : a, b, c \in A\}$$ (Ordered Triplet).

A relation $$R$$ from a non-empty set $$A$$ to a non-empty set $$B$$ is a subset of the cartesian product $$A \times B$$.

Domain	The set of all first elements of the ordered pairs in a relation $$R$$ from a set $$A$$ to a set $$B$$.
Range	The set of all second elements in a relation $$R$$ from a set $$A$$ to a set $$B$$.
Codomain	The whole set $$B$$ is called the codomain of the relation $$R$$. Note that $$Range \subseteq Codomain$$.
Total Relations	If $$n(A) = p$$ and $$n(B) = q$$, then total number of relations is $$2^{pq}$$.

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 $$B$$.

Notation	If $$f$$ is a function from $$A$$ to $$B$$ and $$(a, b) \in f$$, then $$f(a) = b$$. Here, $$b$$ is the image of $$a$$, and $$a$$ is the pre-image of $$b$$.
Real Valued Function	A function which has either $$R$$ or one of its subsets as its range.
Real Function	A function whose domain and range both are either $$R$$ or a subset of $$R$$.

Type	Definition	Characteristics
Identity Function	$$y = f(x) = x$$ for each $$x \in R$$.	Domain: $$R$$, Range: $$R$$. Graph is a straight line passing through origin.
Constant Function	$$y = f(x) = c$$, where $$c$$ is a constant.	Domain: $$R$$, Range: $$\{c\}$$. Graph is parallel to x-axis.
Polynomial Function	$$y = f(x) = a_0 + a_1x + … + a_nx^n$$.	Defined for all $$x \in R$$. Examples: $$x^2, x^3 – x^2 + 2$$.
Rational Function	$$f(x) = \frac{g(x)}{h(x)}$$, where $$g(x), h(x)$$ are polynomials and $$h(x) \neq 0$$.	Domain: $$R – \{x : h(x) = 0\}$$.
Modulus Function	$$f(x) = \|x\| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$$	Domain: $$R$$, Range: $$[0, \infty)$$ (Non-negative real numbers).
Signum Function	$$f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$$	Domain: $$R$$, Range: $$\{-1, 0, 1\}$$.
Greatest Integer Function	$$f(x) = [x]$$, assumes the value of greatest integer less than or equal to $$x$$.	Domain: $$R$$, Range: $$Z$$ (Integers). Example: $$[2.3] = 2, [-1.4] = -2$$.

Let $$f: X \to R$$ and $$g: X \to R$$ be two real functions where $$X \subset R$$.

Addition	$$(f + g)(x) = f(x) + g(x)$$, for all $$x \in X$$.
Subtraction	$$(f – g)(x) = f(x) – g(x)$$, for all $$x \in X$$.
Multiplication by Scalar	$$(\alpha f)(x) = \alpha f(x)$$, $$x \in X$$.
Multiplication of two functions	$$(fg)(x) = f(x)g(x)$$, for all $$x \in X$$.
Quotient	$$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$$, provided $$g(x) \neq 0$$.