Class 11 — Mathematics (NCERT)
Relations and Functions: Cartesian Product and Types of Functions
Class 11
- ✓By the end of this lesson students will be able to define and compute the Cartesian product of two sets.
- ✓By the end of this lesson students will be able to identify and distinguish between relations and functions.
- ✓By the end of this lesson students will be able to understand and apply the concepts of domain, codomain, and range of a function.
- ✓By the end of this lesson students will be able to classify functions as one-one (injective), onto (surjective), many-one, into, or bijective.
- ✓By the end of this lesson students will be able to solve problems involving the properties and types of functions.
Key concepts
The Cartesian product of two non-empty sets A and B is the set of all ordered pairs (a, b) such that 'a' is an element of set A and 'b' is an element of set B. It is denoted by A × B. The order of elements in an ordered pair is important, i.e., (a, b) ≠ (b, a) unless a = b.
A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs.
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. This means two conditions must be satisfied: \n1. Every element in the domain A must be mapped to an element in the codomain B. \n2. Each element in the domain A must be mapped to a unique element in the codomain B (i.e., no element in A has more than one image). We write f: A → B.
For a function f: A → B:\n- The set A is called the domain of the function.\n- The set B is called the codomain of the function.\n- The set of all images of elements of A under f is called the range of the function. The range is always a subset of the codomain (Range ⊆ Codomain).
A function f: A → B is said to be one-one (or injective) if distinct elements of A have distinct images in B. In other words, if f(x₁) = f(x₂) for x₁, x₂ ∈ A, then x₁ = x₂.
A function f: A → B is said to be onto (or surjective) if every element of B is the image of some element of A under f. This means that for every y ∈ B, there exists at least one x ∈ A such that f(x) = y. Equivalently, the range of f is equal to its codomain (Range = Codomain).
A function f: A → B is said to be bijective if it is both one-one (injective) and onto (surjective).
A function f: A → B is said to be many-one if two or more distinct elements of A have the same image in B. It is the negation of a one-one function.
A function f: A → B is said to be an into function if there exists at least one element in B which is not the image of any element of A. In other words, the range of f is a proper subset of its codomain (Range ⊂ Codomain). It is the negation of an onto function.
Key facts to remember
- 1The Cartesian product A × B consists of all ordered pairs (a, b) where a ∈ A and b ∈ B.
- 2If n(A) = p and n(B) = q, then n(A × B) = pq.
- 3A × B ≠ B × A unless A = B or one of the sets is empty.
- 4A relation R from A to B is any subset of A × B.
- 5A function f: A → B is a special type of relation where every element of A has one and only one image in B.
- 6A function is one-one (injective) if distinct elements of the domain have distinct images.
- 7A function is onto (surjective) if its range is equal to its codomain.
- 8A function is bijective if it is both one-one and onto.
Worked examples
Example 1
If A = {1, 2, 3} and B = {a, b}, find A × B and B × A. Also, find the number of elements in A × B and B × A.
Answer
A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}\nB × A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}\nn(A × B) = 6, n(B × A) = 6
Observe that A × B ≠ B × A, although n(A × B) = n(B × A).
Example 2
Let A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. Which of the following relations from A to B are functions?\n(i) R₁ = {(1, 2), (2, 4), (3, 6), (4, 8)}\n(ii) R₂ = {(1, 2), (1, 4), (2, 6), (3, 8)}\n(iii) R₃ = {(1, 2), (2, 4), (3, 6)}
Answer
(i) R₁ is a function.\n(ii) R₂ is not a function.\n(iii) R₃ is not a function.
Always check both conditions for a function: every element of the domain is mapped, and each element has a unique image.
Example 3
Show that the function f: N → N, given by f(x) = 2x, is one-one but not onto. (N is the set of natural numbers).
Answer
The function f: N → N, f(x) = 2x, is one-one but not onto. Hence proved.
For 'not onto' proofs, it is sufficient to find just one element in the codomain that has no pre-image in the domain.
Common mistakes
- ✗Confusing an ordered pair (a, b) with a set {a, b} or assuming (a, b) is the same as (b, a).
- ✗Failing to check both conditions for a function: that every element in the domain is mapped, and that each element has a unique image.
- ✗Incorrectly assuming that if a function is not one-one, it must be onto, or vice-versa.
- ✗Not clearly distinguishing between the codomain and the range of a function.
- ✗Making errors in algebraic manipulation when proving one-one or onto properties for functions defined by rules.
Exam tips
- ★Always clearly state the domain and codomain of the function at the beginning of your solution.
- ★For proving a function is one-one, assume f(x₁) = f(x₂) and logically deduce x₁ = x₂.
- ★For proving a function is onto, take an arbitrary element 'y' from the codomain and show that there exists an 'x' in the domain such that f(x) = y.
- ★For small finite sets, drawing arrow diagrams can help visualise relations and functions and determine their types.
- ★Practice with various types of functions (polynomial, rational, trigonometric, exponential, logarithmic) to understand how their properties affect injectivity and surjectivity.
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