# Relations and Functions | Notes | Math XI

# Relations and Functions

## Key Concepts

- A pair of elements grouped together in a particular order is known as an
**ordered pair**. - The two ordered pairs (a, b) and (c, d) are said to be equal if and only if a = c and b = d.
- Let A and B be any two non-empty sets.

The**Cartesian product**A × B is the set of all ordered pairs of elements of sets from A and B defined as follows:

**A × B = {(a, b): a ∈ A, b ∈ B}**.

Cartesian product of two sets is also known as the**product set**. - If any of the sets of A or B or both are empty, then the set A × B will also be empty and consequently,
**n(A × B) = 0**. - If the number of elements in A is m and the number of elements in set B is n, then the set A × B will have mn elements.
- If any of the sets A or B is infinite, then A × B is also an
**infinite set**. - Cartesian product of sets can be extended to three or more sets. If A, B and C are three non-empty sets, then
**A × B × C = {(a, b, c): a ∈ A, b ∈ B, c ∈ C}**. Here (a, b, c) is known as an**ordered triplet**. - Cartesian product of a non-empty set A with an empty set is an
**empty set**, i.e.**A × φ = φ**. - The Cartesian product is not commutative, namely A × B is not the same as B × A, unless A and B are equal.
- The Cartesian product is associative, namely
**A × (B × C) = (A × B) × C** - R × R = {(a, b): a ∈ R, b ∈ R} represents the coordinates of all points in the two-dimensional plane.
**R × R × R = {(a, b, c): a ∈ R, b ∈ R, c ∈ C}**represents the coordinates of all points in the three-dimensional plane. - A
**relation**R from a non-empty set A to another non-empty set B is a subset of their Cartesian product A × B, i.e. R ⊆ A × B. - If (x, y) ∈ R or x R y, then ‘x’ is related to ‘y’.
- If (x, y) ∉ R or , then ‘x’ is not related to ‘y’.
- The second element b in the ordered pair (a, b) is the
**image**of the first element a, and a is the**preimage**of b. - The
**domain**of R is the set of all first elements of the ordered pairs in a relation R. In other words, the domain is the set of all the inputs of the relation. - If the relation R is from a non-empty set A to non-empty set B, then set B is called the
**co-domain**of the relation R. - The set of all the images or the second element in the ordered pair (a, b) of relation R is called the
**range**of R. - The total number of relations which can be defined from a set A to a set B is the number of possible subsets of A × B.
- A × B can have 2
^{mn}subsets. This means there are 2^{mn}relations from A to B. - Relation can be represented algebraically and graphically. The various methods of representation are as follows:

- A relation ‘f’ from a non-empty set A to another non-empty set B is said to be a
**function**if every element of A has a unique image in B. - The domain of ‘f’ is the set A. No two distinct ordered pairs in ‘f’ have the same first element.
- Every function is a relation but the converse is not true.
- 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**pre-image**of b under f. - If f: A → B, then A is the domain and B is the co-domain of f.
- The range of the function is the set of images.
- A
**real function**has the set of real numbers or one of its subsets both as its domain and as its range. - Identity function: f: X → X is an identity function if f(x) = x for each x ∈ A

- Graph of the identity function is a straight line which makes an angle of 45° with both the x and the y-axis, respectively. All points on this line have their x and y coordinates equal.
**Constant function:**A constant function is one which maps each element of the domain to a constant.

Domain of this function is R and range is the singleton set {c}, where c is a constant.

- Graph of a constant function is a line parallel to the x-axis. The graph lies above the x-axis if the constant c > 0, below the x-axis if the constant c < 0 and is the same as the x-axis if c = 0.
**Polynomial function:**f: R → R defined as y = f(x) = a_{0}+ a_{1}x + a_{2}x^{2}+ …. + a_{n}x^{n}, where n is a nonnegative integer and a_{0}, a_{1}, a_{2}, …a_{n}∈ R.- A linear polynomial represents a straight line, while a quadratic polynomial represents a parabola.
- Functions of the form , where f(x) and g(x) ≠ 0 are polynomial functions, are called rational functions.
- Domain of rational functions does not include those points where g(x) = 0. For example, the domain of is R – {2}.
**Modulus function**: f: R → R denoted by f(x) = |x| for each x ∈ R.

The modulus function is defined as f(x) = x if x ≥ 0 and f(x) = –x if x < 0. The graph of a modulus function is above the x-axis as shown in the figure.

**Step or greatest integer function**: A function f: R → R denoted by f(x) = [x], x ∈ R, where [x] is the value of greatest integer, less than or equal to ‘x’ is called a step or greatest integer function. It is also called a**floor function.**

**Smallest integer function**: A function f: R → R denoted by f(x) = ⌈x⌉, x ∈ R where ⌈x⌉ is the value of the smallest integer, greater than or equal to ‘x’ is called a**smallest integer function**. It is also known as the**ceiling function.**

**Signum function**: , x ≠ 0 and 0 for x = 0. The domain of a signum function is R and its range is {–1, 0, 1}.

- If ‘a’ is a positive real number other than unity, then a function which relates each x ∈ R to a
^{x}is called the**exponential function**.

- If a > 0 and a ≠ 1, then the function defined by
**f(x) = log**, x > 0 is called the_{a}x**logarithmic function**.

- The function defined by f :R – {0} → R such that, f(x) =1/x is called the
**reciprocal**function.

- The function defined by f :R
^{+}→ R such that, f(x) = +**√**x is called the**square root function**.

- The function defined by f :R → R such that, f(x) = x
^{2}is called the**square function**.

- The function defined by f :R → R such that f(x) = x
^{3}is called the**cube function**.

- The function defined by f :R → R such that, f(x) = ∛x is called the
**cube root function**.

## Key Formulae

- 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.
- (a, b, c) = (d, e, f) if a = d, b = e and c = f.
- If n(A) = n and n(B) = m, then n(A × B) = mn.
- If n(A) = n and n(B) = m, then 2
^{mn}relations can be defined from A to B. - Algebra of Real function:

For function f : X → R and g: X → R, we have- (f + g) (x) = f(x) + g(x), x ∈ X.
- (f – g) (x) = f(x) – g(x), x ∈ X.
- (f.g) (x) = f(x). g(x), x ∈ X.
- (kf) (x) = kf(x), x ∈ X, where k is a real number.
- , x ∈ X., g(x) ≠ 0.