2.1 Cartesian Product of Sets
Given two non-empty sets P and Q, the Cartesian product P × Q is the set of all ordered pairs of elements from P and Q:
P × Q = {(p, q) : p ∈ P, q ∈ Q}
If either P or Q is the null set, then P × Q = φ.
[ Image: Arrow diagram of Cartesian product (e.g., colours × objects) — add here ]
Key properties of Cartesian products:
If n(A) = p and n(B) = q, then n(A × B) = pq
In general, A × B ≠ B × A (order of the pair matters)
R × R = set of all coordinate points in 2D plane; R × R × R = all points in 3D space
A × A × A = {(a, b, c) : a, b, c ∈ A} — called an ordered triplet
Example: If A = {1, 2} and B = {3, 4}, then A × B = {(1,3), (1,4), (2,3), (2,4)} → n(A × B) = 4
🎯 Practice MCQs
n(A) = 3, n(B) = 3 (elements 3, 4, 5). Using the formula n(A × B) = n(A) × n(B) = 3 × 3 = 9.
A = set of first elements = {p, m} → n(A) = 2. B = set of second elements = {q, r} → n(B) = 2. And indeed n(A × B) = 4 = 2 × 2 ✓
Corresponding elements of equal ordered pairs are equal:
x + 1 = 3 → x = 2
y – 2 = 1 → y = 3
2.2 Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. It is obtained by describing a relationship between the first and second elements of ordered pairs in A × B.
The domain of R = set of all first elements.
The range of R = set of all second elements.
The whole set B = codomain of R. Note: Range ⊆ Codomain.
[ Image: Arrow diagram showing a relation between two sets — add here ]
If n(A) = p and n(B) = q, then the total number of possible relations from A to B = 2pq.
Let A = {1,2,3,4,5,6}. R = {(x,y) : y = x+1} = {(1,2),(2,3),(3,4),(4,5),(5,6)}
Domain = {1,2,3,4,5}, Range = {2,3,4,5,6}, Codomain = {1,2,3,4,5,6}
🎯 Practice MCQs
n(A) = 2, n(B) = 2. So n(A × B) = 4. Total relations = 2^(n(A×B)) = 2^4 = 16.
R = {(1,2),(2,3),(3,4),(4,5),(5,6)}. First elements (domain) = {1,2,3,4,5}. Second elements (range) = {2,3,4,5,6}. Element 6 is in A but has no image (6+1=7 ∉ A), so 6 ∉ domain.
Prime numbers less than 10 are: 2, 3, 5, 7. These are the first elements (x values) of the ordered pairs — the domain. The range would be {8, 27, 125, 343}.
2.3 Functions
A relation f from set A to set B is called a function if every element of set A has one and only one image in set B.
Written as f : A → B, where f(a) = b means b is the image of a, and a is the preimage of b.
A is the domain, B is the codomain, and the set of all images is the range.
[ Image: Arrow diagram comparing function vs non-function — add here ]
When is a relation NOT a function?
If any element of domain has no image in B (e.g., R = {(1,2),(2,3)} on A = {1,2,3} — element 3 has no image)
If any element of domain has more than one image (e.g., R = {(2,2),(2,4),(3,3)} — element 2 maps to both 2 and 4)
A real valued function has R or its subset as range. A real function also has R or its subset as domain.
Function f(x) = 2x + 1 for x = 1 to 7:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| f(x) = 2x+1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
🎯 Practice MCQs
In B, each element has exactly one image: 1→2, 2→3, 3→4, 4→5, 5→6, 6→7 — it is a function.
A: element 2 maps to both 2 and 4 (not a function).
C: element 1 maps to both 3 and 5 (not a function).
D: element 3 has no image (not a function).
f(7) = 2(7) – 5 = 14 – 5 = 9.
Denominator x²–5x+4 = (x–1)(x–4). Function is undefined when denominator = 0, i.e., at x = 1 and x = 4. So domain = R – {1, 4}.
2.4 Types of Functions and Their Graphs
Seven important types of real functions with their key properties:
f(x) = x for all x ∈ R. Domain = R, Range = R. Graph: straight line through origin with slope 1.
f(x) = c (constant) for all x ∈ R. Domain = R, Range = {c}. Graph: horizontal line parallel to x-axis.
f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ where n is a non-negative integer and aᵢ ∈ R. Examples: f(x) = x² (parabola), f(x) = x³ (cubic).
f(x) = p(x)/q(x) where p(x) and g(x) are polynomials and g(x) ≠ 0. Example: f(x) = 1/x, Domain = R–{0}, Range = R–{0}.
f(x) = |x| = x if x ≥ 0; –x if x < 0. Domain = R, Range = [0, ∞). V-shaped graph.
f(x) = 1 if x > 0; 0 if x = 0; –1 if x < 0. Domain = R, Range = {–1, 0, 1}.
f(x) = [x] = greatest integer ≤ x. Also called floor function. Example: [2.7] = 2, [–1.3] = –2. Graph: step function.
[ Image: Graphs of identity, modulus, signum, and greatest integer functions — add here ]
🎯 Practice MCQs
The signum function outputs exactly three values: 1 for positive x, 0 for x=0, and –1 for negative x. So Range = {–1, 0, 1}.
[x] is the greatest integer LESS THAN OR EQUAL to x. The greatest integer ≤ –2.5 is –3 (since –3 < –2.5 < –2, the greatest integer not exceeding –2.5 is –3).
f(x) = 1/x is undefined at x = 0, so domain = R – {0}. For any non-zero y, x = 1/y gives a valid input — so f(x) takes all values except 0. Range = R – {0}.
2.5 Algebra of Real Functions
For functions f : X → R and g : X → R, we define the following operations:
Sum: (f + g)(x) = f(x) + g(x), for all x ∈ X
Difference: (f – g)(x) = f(x) – g(x), for all x ∈ X
Product: (fg)(x) = f(x) · g(x), for all x ∈ X (pointwise multiplication)
Scalar multiple: (kf)(x) = k · f(x), for all x ∈ X, where k is a real number
Quotient: (f/g)(x) = f(x)/g(x), for all x ∈ X where g(x) ≠ 0
[ Image: Example illustration of function operations on graphs — add here ]
Example: If f(x) = x² and g(x) = 2x + 1:
🎯 Practice MCQs
(f + g)(x) = x² + 2x + 1
(f + g)(2) = 4 + 4 + 1 = 9.
Alternatively: f(2) + g(2) = 4 + 5 = 9 ✓
(fg)(x) = f(x) · g(x) = √x · x = x · x^(1/2) = x^(1 + 1/2) = x^(3/2).
(f/g)(x) = f(x)/g(x) = (x+1)/(2x–3).
(f/g)(2) = (2+1)/(2·2–3) = 3/(4–3) = 3/1 = 3.
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