CBSE Class 11 Maths Chapter 2 Notes: Relations and Functions

Cartesian Product of Sets and Relations

The concept of Cartesian Product is fundamental to understanding relations. If A and B are two non-empty sets, the Cartesian product A × B is the set of all ordered pairs (a, b) such that 'a' belongs to set A and 'b' belongs to set B. Mathematically, A × B = {(a, b) : a ∈ A, b ∈ B}. The order of elements in an ordered pair matters, i.e., (a, b) ≠ (b, a) unless a = b. If either A or B is an empty set, then A × B is also an empty set. If n(A) = p and n(B) = q, then the number of elements in A × B is n(A × B) = pq.

A Relation R from a non-empty set A to a non-empty set B is defined as a subset of the Cartesian product A × B. This means that for any pair (a, b) to be in the relation R, 'a' must be from set A and 'b' from set B, and they must satisfy a specific rule or condition. The Domain of a relation R is the set of all first elements (x-coordinates) of the ordered pairs in R. The Codomain of a relation R from A to B is the entire set B. The Range of a relation R is the set of all second elements (y-coordinates) of the ordered pairs in R. It's important to note that the Range is always a subset of the Codomain. Relations can be represented using roster form, set-builder form, or arrow diagrams.

Key Definitions

Cartesian Product

For non-empty sets A and B, A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Ordered Pair

A pair of elements (a, b) where the order of elements is significant. (a, b) ≠ (b, a) unless a=b.

Relation

A subset of the Cartesian product A × B, defined by some property or rule between the elements of A and B.

Domain of a Relation

The set of all first elements of the ordered pairs in a relation R.

Codomain of a Relation

If R is a relation from A to B, then B is the codomain.

Range of a Relation

The set of all second elements of the ordered pairs in a relation R. It is a subset of the codomain.

Function

A special type of relation from set A to set B where every element of set A has one and only one image in set B.

Real Function

A function whose domain and codomain are subsets of the set of real numbers (R).

Functions: A Special Type of Relation

A function is a specific kind of relation. A relation f from a set A to a set B is called a function if every element of set A has one and only one image in set B. This means two conditions must be met:

1. Every element in the domain must be mapped: For every x ∈ A, there exists some y ∈ B such that (x, y) ∈ f.
2. Uniqueness of image: If (x, y₁) ∈ f and (x, y₂) ∈ f, then y₁ = y₂. In simpler terms, an element in A cannot be related to more than one element in B.

We typically denote a function as f: A → B, where A is the domain and B is the codomain. The range of a function f is the set of all images of the elements of A under f. If the domain and codomain are subsets of real numbers, it's called a real function.

Common Types of Real Functions:

• Identity Function: f(x) = x
• Constant Function: f(x) = c (where c is a constant)
• Polynomial Functions: f(x) = a₀ + a₁x + ... + aₙxⁿ
• Rational Functions: f(x) = P(x)/Q(x) where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0.
• Modulus Function (Absolute Value Function): f(x) = |x| = {x if x ≥ 0, -x if x < 0}
• Signum Function: f(x) = {1 if x > 0, 0 if x = 0, -1 if x < 0}
• Greatest Integer Function (Step Function): f(x) = [x], which gives the greatest integer less than or equal to x.

Algebra of Real Functions: For functions f: X → R and g: X → R, we can define:

• (f + g)(x) = f(x) + g(x)
• (f - g)(x) = f(x) - g(x)
• (fg)(x) = f(x)g(x)
• (f/g)(x) = f(x)/g(x), provided g(x) ≠ 0.

Relation vs. Function

Worked Examples

• {"title":"Cartesian Product","description":"If set A = {1, 2} and set B = {a, b}, find A × B and B × A.","solution":"A × B = {(1, a), (1, b), (2, a), (2, b)}\nB × A = {(a, 1), (a, 2), (b, 1), (b, 2)}"}
• {"title":"Identifying a Function","description":"Let A = {1, 2, 3} and B = {p, q}. Which of the following relations is a function from A to B?\n(i) R₁ = {(1, p), (2, q), (3, p)}\n(ii) R₂ = {(1, p), (1, q), (2, p), (3, q)}","solution":"(i) R₁ is a function because every element of A has exactly one image in B. (1 maps to p, 2 to q, 3 to p).\n(ii) R₂ is not a function because element '1' from set A has two images ('p' and 'q') in set B. Also, element '3' is mapped to 'q' but '2' is mapped to 'p', and '1' is mapped twice. The rule is violated."}

Exam Tip: Domain and Range Pitfalls

When finding the domain of a real function, always look for values of 'x' that would make the function undefined. Common pitfalls include:

1. Denominators: The denominator cannot be zero. For f(x) = 1/Q(x), ensure Q(x) ≠ 0.
2. Square Roots: The expression under an even root (like square root) cannot be negative. For f(x) = √P(x), ensure P(x) ≥ 0.

For range, try to express 'x' in terms of 'y' (if y=f(x)) and then find the valid values for 'y'. Practice with various function types, especially modulus and greatest integer functions, as they often trick students.

Key Points to Remember

• The order of elements in an ordered pair (a, b) matters; (a, b) ≠ (b, a) unless a = b.
• If n(A) = p and n(B) = q, then n(A × B) = pq.
• A relation from A to B is always a subset of A × B.
• A function is a special type of relation where every element in the domain (A) has one and only one image in the codomain (B).
• The range of a relation/function is always a subset of its codomain.
• For a function f: A → B, A is the domain, B is the codomain, and {f(x) | x ∈ A} is the range.
• The vertical line test can be used to graphically determine if a relation is a function: if any vertical line intersects the graph at more than one point, it is not a function.
• Pay close attention to inequalities when determining the domain for square root functions or rational functions.

Practice Questions with Solutions

• Q1: If A = {x, y, z} and B = {1, 2}, how many elements are in A × B? List them. A1: n(A × B) = n(A) × n(B) = 3 × 2 = 6 elements. A × B = {(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)}.
• Q2: Can the range of a function be equal to its codomain? Give an example. A2: Yes, the range can be equal to the codomain. For example, for f: R → R defined by f(x) = x (identity function), the range is R, which is equal to the codomain.
• Q3: For the function f(x) = 1/(x-3), find its domain. A3: The function is defined for all real numbers except where the denominator is zero. So, x - 3 ≠ 0 ⇒ x ≠ 3. The domain is R - {3} or (-∞, 3) ∪ (3, ∞).
• Q4: A relation R = {(1, 2), (2, 3), (1, 4)} is given. Is R a function? Why or why not? A4: No, R is not a function. The element '1' from the domain is mapped to two different elements ('2' and '4') in the codomain, violating the uniqueness condition of a function.

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