Relations and Functions - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter - 2
RELATIONS AND FUNCTIONS

---

1. Cartesian Product of Sets
2. Relations
3. Functions

•  Ordered pair  A pair of elements grouped together in a particular order. Clearly, (a,b)≠(b,a)$\left(a,b\right)\ne \left(b,a\right)$.

• Cartesian product of two sets A and B is given by A × B =  {(a, b): a ∈ A, b ∈ B}.

In particular R × R = {(x, y): x, y ∈ R} and R × R × R = (x, y, z): x, y, z ∈ R}

• If (a, b) = (x, y), then a =  x and b = y.
•  If n(A) = p and  n(B) =  q, then n(A × B) = pq.
• A × φ = φ
•  In general, A × B ≠ B × A.
•  Relation: Relation A relation R from a set A to a set B is a subset of the Cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B, i.e., R⊆A×B$\mathrm{R}\subseteq \mathrm{A}\times \mathrm{B}$.
• Number of Relations: Let A and B be two non-empty finite sets, comtaining m and n elements respectively, then the total number of relaitons from A to B is 2mn$2^{mn}$
• Domain: The domain of R is the set of all first elements of the ordered pairs in a relation R. Domain R = {a:(a,b)∈R}$\left\{a:\left(a,b\right)\in \mathrm{R}\right\}$.

• The image of an element x under a relation R is given by y, where (x, y) ∈ R,
• Range: The range of the relation R is the set of all second elements of the ordered pairs in a relation R. Range R = {b:(a,b)∈R}$\left\{b:\left(a,b\right)\in \mathrm{R}\right\}$.
• Function: Function A function f  from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write  f: A→B, where f(x) =  y.
• Domain and Co-domain: The set A is called the domain of function f and the set B is called the co-domain of f.
• Range: If f is a function from A to B, then each element of A corresponds to ine and only one element of B, whereas every element in B need not be the image of some x$x$ in A. The subset of B comtaining the image of elements of A is called the range of the function. The range of f is denoted by f(A)$f\left(\mathrm{A}\right)$. Mathematically, we write: f(A)={f(x):x∈A}$f\left(\mathrm{A}\right)=\left\{f\left(x\right):x\in \mathrm{A}\right\}$
• Image: If the element x of A corresponds to y∈B$y\in \mathrm{B}$ under the function f, then we say that y$y$ is the image of x$x$ under f and we write, f(x)=y$f\left(x\right)=y$.
• Pre-image: If f(x)=y$f\left(x\right)=y$, then x$x$ is pre-image of y.