CBSE Class 11 Maths Chapter 2: Relations And Functions Notes

Cartesian Product of Sets

The concept of Relations and Functions starts with the fundamental pairing of sets. Let $A$ and $B$ be two non-empty sets. The Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$.

Mathematically, it is written as:
$A \times B = \{(a, b) : a \in A \text{ and } b \in B\}$

If either $A$ or $B$ is an empty set ($\emptyset$), then $A \times B = \emptyset$. If either $A$ or $B$ is infinite, then $A \times B$ is also infinite. An ordered pair $(a, b)$ is not equal to $(b, a)$ unless $a = b$. Ordered pairs are equal if and only if their corresponding first elements and second elements are respectively equal: $(x, y) = (a, b) \iff x = a \text{ and } y = b$.

Essential Definitions & Terminology

Cartesian Product

The collection of all possible ordered pairs where the first element is chosen from set A and the second element is chosen from set B.

Relation (R)

A subset of the Cartesian product A × B. It expresses a relationship between the first element 'x' and the second element 'y' of the ordered pair.

Domain

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

Range

The set of all second elements of the ordered pairs in a relation or a function that have an active mapping (actual output values).

Codomain

The entire second set B in the mapping f: A → B. Range is always a subset of Codomain (Range ⊆ Codomain).

Function

A special type of relation in which every element of the domain has one, and only one, unique image in the codomain.

Modulus Function

A real function defined by f(x) = |x|, yielding x for x ≥ 0, and -x for x < 0. Its graph is V-shaped.

Greatest Integer Function

A step function f(x) = [x], which rounds down any real number x to the greatest integer less than or equal to x.

Comparison: Relation vs. Function

Key Points & Formulas (Must-Remember)

• If the number of elements in Set A is p, i.e., n(A) = p, and in Set B is q, i.e., n(B) = q, then n(A × B) = pq.
• The total number of possible relations from Set A to Set B is 2^(pq).
• The range is always a subset of the codomain: Range ⊆ Codomain.
• A relation from A to B is defined as a function f if and only if Domain of f = A, and no two distinct ordered pairs in f have the same first coordinate.
• Algebra of Real Functions: (f + g)(x) = f(x) + g(x); (f - g)(x) = f(x) - g(x); (fg)(x) = f(x)g(x).
• For division, (f/g)(x) = f(x) / g(x), provided g(x) ≠ 0.
• Domain of (f + g), (f - g), and (fg) is the intersection of their individual domains: D_f ∩ D_g.
• Signum Function value is always: 1 if x > 0, 0 if x = 0, and -1 if x < 0. Domain is R, Range is {-1, 0, 1}.

Step-by-Step: Finding the Domain and Range of Real Functions

1. Step 1: Check for Rational Expressions (Division by Zero) — If the function contains a fraction, identify any values of x that make the denominator zero. Exclude these values from the domain. (e.g., for 1/(x-2), x ≠ 2).
2. Step 2: Check for Even Roots (Radical Expressions) — If there is a square root (or any even root) in the numerator, set the expression inside the radical to be greater than or equal to zero (expression ≥ 0) and solve for x.
3. Step 3: Combine Constraints — Find the intersection of all conditions generated in Step 1 and Step 2 to obtain the absolute Domain of the function.
4. Step 4: Find the Range algebraically — Express x in terms of y (i.e., x = g(y)). Find all valid values of y for which x is defined as a real number under the domain constraints calculated.

Worked-Out Examples

• {"title":"Example 1: Elements of Cartesian Product","bodyMarkdown":"Question: If $A = \\{1, 2, 3\\}$ and $B = \\{3, 4\\}$, find $A \\times B$ and determine the total number of relations possible from $A$ to $B$.\n\nSolution:\n1. Find the elements of $A \\times B$ by pairing every element of $A$ with every element of $B$:\n $A \\times B = \\{(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)\\}$\n2. Calculate the number of elements:\n $n(A) = 3$ and $n(B) = 2 \\implies n(A \\times B) = 3 \\times 2 = 6$\n3. Calculate the total number of relations:\n Number of relations $= 2^{n(A \\times B)} = 2^6 = 64$."}
• {"title":"Example 2: Finding Domain and Range of a Radical Function","bodyMarkdown":"Question: Find the domain and range of the real function $f(x) = \\sqrt{9 - x^2}$.\n\nSolution:\n1. For Domain: The value inside the square root must be non-negative.\n $9 - x^2 \\ge 0 \\implies x^2 \\le 9 \\implies -3 \\le x \\le 3$\n Therefore, Domain $= [-3, 3]$.\n2. For Range: Let $y = \\sqrt{9 - x^2}$. Since square roots are non-negative, $y \\ge 0$.\n Squaring both sides: $y^2 = 9 - x^2 \\implies x^2 = 9 - y^2$\n For $x$ to be a real number, $9 - y^2 \\ge 0 \\implies y^2 \\le 9 \\implies -3 \\le y \\le 3$.\n Combining with the condition $y \\ge 0$, we get $0 \\le y \\le 3$.\n Therefore, Range $= [0, 3]$."}

Board Exam Tips & Common Traps

• The Common Interval Trap: Students often confuse closed brackets [a, b] (which include endpoints) with open brackets (a, b) (which exclude endpoints) when writing domains/ranges. Double-check your endpoints!
• Is it a Function? Check the inputs! If a question gives you a set of ordered pairs and asks if it is a function, look strictly at the first elements. If any first element repeats with a different second element, write down immediately: "Since the element X has more than one image, this relation is not a function." This earns full step-marking.
• Domain Intersection: When adding, subtracting, or multiplying two functions, remember that the resulting domain is strictly $D_f \cap D_g$. Do not just blindly write the domain of the simplified final algebraic expression.

Practice Questions with Solutions

• If A × B = {(a, x), (a, y), (b, x), (b, y)}, find sets A and B. Set A is the set of first coordinates: A = {a, b}. Set B is the set of second coordinates: B = {x, y}.
• What is the domain of the real function f(x) = 1 / (x^2 - 4)? The denominator must not be zero: x^2 - 4 ≠ 0 => x^2 ≠ 4 => x ≠ 2 and x ≠ -2. Therefore, Domain = R - {-2, 2}.
• If R is a relation on Set A = {1, 2, 3} defined by R = {(1,1), (2,2), (3,3), (1,2)}, is R a function? No. Here, the element 1 has two different images: 1 and 2 (as indicated by the pairs (1,1) and (1,2)). Thus, it is not a function.
• State the range of the Modulus Function f(x) = |x| defined on R. The Modulus function always outputs a non-negative real value. Hence, Range = [0, ∞) or the set of non-negative real numbers.

Frequently Asked Questions