# 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 $\inline x \not \mathrel{R} y$, 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 2mn subsets. This means there are 2mn 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) = a0 + a1x + a2x2 + …. + an xn, where n is a nonnegative integer and a0, a1, a2, …an ∈ R.
• A linear polynomial represents a straight line, while a quadratic polynomial represents a parabola.
• Functions of the form $\mathrm{\frac{f(x)}{g(x)}}$, 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 $\mathrm{f(x)=\frac{1}{x-2}}$ 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: $\mathrm{ f(x)=\frac{|x|}{x}}$ , 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 ax is called the exponential function.
• If a > 0 and a ≠ 1, then the function defined by f(x) = loga x, x > 0 is called the 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) = x2 is called the square function.
• The function defined by f :R → R such that f(x) = x3 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 2mn 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.
• $\mathrm{\left (\frac{f}{g }\right )(x)=\frac{f(x)}{g(x)}}$, x ∈ X., g(x) ≠ 0.