Home Class 11 Applied Maths Chapter 5: Relations Exercise 5.1: Cartesian Product of Sets
Chapter 5 • Unit II

Exercise 5.1: Cartesian Product of Sets

Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).

5.1 Cartesian Product of Sets

In Applied Mathematics, relational databases, inventory modeling, and algorithmic design, the Cartesian Product is the fundamental operation that constructs multi-dimensional structures from one-dimensional collections. It provides the rigorous formal framework for product variants (SKUs), coordinates in plane geometry, and pairing dependencies.

Core Structural Theorems & Definitions

  • Ordered Pair: An ordered pair \((a, b)\) consists of two elements \(a\) and \(b\) in a specified order, where \(a\) is the first coordinate and \(b\) is the second coordinate. \[ (a_1, b_1) = (a_2, b_2) \iff a_1 = a_2 \quad \text{and} \quad b_1 = b_2 \]
  • Cartesian Product (\(A \times B\)): Given two non-empty sets \(A\) and \(B\), the Cartesian product \(A \times B\) is the set of all ordered pairs \((a, b)\) such that \(a \in A\) and \(b \in B\): \[ A \times B = \{ (a, b) : a \in A \text{ and } b \in B \} \]
  • Cardinality Theorem: If \(n(A) = p\) and \(n(B) = q\), then: \[ n(A \times B) = n(A) \times n(B) = p \cdot q \]
  • Non-Commutative Property: \[ A \times B \neq B \times A \quad (\text{unless } A = B \text{ or } A = \emptyset \text{ or } B = \emptyset) \]
  • Empty Set Identity: If either \(A = \emptyset\) or \(B = \emptyset\), then: \[ A \times B = \emptyset \]
  • Distributive Identities:
    1. \(A \times (B \cup C) = (A \times B) \cup (A \times C)\)
    2. \(A \times (B \cap C) = (A \times B) \cap (A \times C)\)
    3. \(A \times (B - C) = (A \times B) - (A \times C)\)
    4. \((A \times B) \cap (S \times T) = (A \cap S) \times (B \cap T)\)

1. Equality of Ordered Pairs: Algebraic Foundations

Two ordered pairs are equal if and only if their respective first components are equal and their respective second components are equal. In CBSE examinations, this is frequently tested as linear or quadratic systems:

Example: Solving for Unknown Parameters

Problem: If \(\left( \frac{x}{3} + 1, \, y - \frac{2}{3} \right) = \left( \frac{5}{3}, \, \frac{1}{3} \right)\), find the values of \(x\) and \(y\).

Solution: Equating corresponding components:

\[ \frac{x}{3} + 1 = \frac{5}{3} \implies \frac{x}{3} = \frac{5}{3} - 1 = \frac{2}{3} \implies x = 2 \] \[ y - \frac{2}{3} = \frac{1}{3} \implies y = \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \]

Therefore, \(x = 2\) and \(y = 1\).

2. Geometric and Real-World Interpretations

The Cartesian product extends into higher dimensions:

  • \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^2\): Represents the set of all coordinates \((x, y)\) on the two-dimensional Cartesian plane.
  • \(\mathbb{R} \times \mathbb{R} \times \mathbb{R} = \mathbb{R}^3\): Represents 3-dimensional space where each point is an ordered triplet \((x, y, z)\).
  • E-Commerce SKU Enumeration: If a retail brand offers T-shirts in 3 sizes \(S = \{\text{S, M, L}\}\) and 4 colors \(C = \{\text{Red, Blue, Black, White}\}\), the total catalog of distinct product SKUs is \(S \times C\), giving \(3 \times 4 = 12\) items.

Example: Intersection of Cartesian Products

Problem: If \(A = \{1, 2\}\), \(B = \{2, 3\}\), and \(C = \{0, 5\}\), verify that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\).

Solution:

  • First evaluate \(B \cap C\): Since \(B = \{2, 3\}\) and \(C = \{0, 5\}\), \(B \cap C = \emptyset\).
  • LHS: \(A \times (B \cap C) = A \times \emptyset = \emptyset\).
  • RHS: \[ A \times B = \{(1, 2), (1, 3), (2, 2), (2, 3)\} \] \[ A \times C = \{(1, 0), (1, 5), (2, 0), (2, 5)\} \] Notice that \(A \times B\) and \(A \times C\) share no common ordered pairs. Therefore, \((A \times B) \cap (A \times C) = \emptyset\).
  • Since LHS \(= \emptyset =\) RHS, the distributive identity is verified.
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