Home Class 11 Applied Maths Chapter 5: Relations Exercise 5.2: Relations
Chapter 5 • Unit II

Exercise 5.2: Relations

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

5.2 Relations: Foundations, Domain, Range, and Co-Domain

In Applied Mathematics, computational systems, and relational databases, a relation establishes structured linkages between distinct entities. From mapping bank customers to credit accounts to tracking logistics routes between suppliers and warehouses, relations translate intuitive real-world dependencies into rigorous set-theoretic algebra.

Core Structural Theorems & Definitions

  • Mathematical Definition of a Relation: A relation \(R\) from a non-empty set \(A\) to a non-empty set \(B\) is defined as any subset of the Cartesian product \(A \times B\): \[ R \subseteq A \times B \] If \((a, b) \in R\), we say that \(a\) is related to \(b\) under \(R\), denoted \(a \, R \, b\).
  • Total Number of Relations: If \(n(A) = p\) and \(n(B) = q\), the Cartesian product contains \(p \cdot q\) ordered pairs. The total number of possible relations is the total number of subsets of \(A \times B\): \[ \text{Total Relations} = 2^{p \cdot q} \] The number of non-empty relations is \(2^{p \cdot q} - 1\).
  • Domain of a Relation: The set of all first coordinates of the ordered pairs belonging to \(R\): \[ \text{Dom}(R) = \{ a \in A : (a, b) \in R \text{ for some } b \in B \} \subseteq A \]
  • Range of a Relation: The set of all second coordinates of the ordered pairs belonging to \(R\): \[ \text{Range}(R) = \{ b \in B : (a, b) \in R \text{ for some } a \in A \} \subseteq B \]
  • Co-Domain: The entire target set \(B\) is called the co-domain of \(R\). Note that: \[ \text{Range}(R) \subseteq \text{Co-Domain}(R) \]
  • Inverse Relation (\(R^{-1}\)): If \(R\) is a relation from \(A\) to \(B\), the inverse relation \(R^{-1}\) is a relation from \(B\) to \(A\) defined by: \[ R^{-1} = \{ (b, a) : (a, b) \in R \} \] where \(\text{Dom}(R^{-1}) = \text{Range}(R)\) and \(\text{Range}(R^{-1}) = \text{Dom}(R)\).

1. Visualizing the Distinction: Range vs Co-Domain

A central conceptual point in CBSE Applied Mathematics is that the Range is the set of actual outputs / matched elements, whereas the Co-Domain is the complete set of potentially eligible outputs.

Example: Determining Domain, Range, and Co-domain

Problem: Let \(A = \{1, 2, 3, 4, 5, 6\}\). Define a relation \(R\) from \(A\) to \(A\) by: \[ R = \{ (x, y) : y = x + 1 \} \]

  1. Write \(R\) in roster form.
  2. Find the Domain, Range, and Co-domain of \(R\).
  3. Write the inverse relation \(R^{-1}\).

Solution:

  1. Evaluating for each \(x \in A\) such that \(y = x + 1 \in A\):
    • For \(x = 1\), \(y = 2 \in A\) \(\implies (1, 2) \in R\)
    • For \(x = 2\), \(y = 3 \in A\) \(\implies (2, 3) \in R\)
    • For \(x = 3\), \(y = 4 \in A\) \(\implies (3, 4) \in R\)
    • For \(x = 4\), \(y = 5 \in A\) \(\implies (4, 5) \in R\)
    • For \(x = 5\), \(y = 6 \in A\) \(\implies (5, 6) \in R\)
    • For \(x = 6\), \(y = 7 \notin A\) (excluded)
    Therefore: \(R = \{ (1, 2), (2, 3), (3, 4), (4, 5), (5, 6) \}\).
  2. \[ \text{Domain}(R) = \{1, 2, 3, 4, 5\} \] \[ \text{Range}(R) = \{2, 3, 4, 5, 6\} \] \[ \text{Co-domain}(R) = A = \{1, 2, 3, 4, 5, 6\} \]
  3. \[ R^{-1} = \{ (2, 1), (3, 2), (4, 3), (5, 4), (6, 5) \} \]

2. Applied Business Context: Relational Modeling

Consider an e-commerce platform with a set of Customers \(C = \{c_1, c_2, c_3\}\) and Product Categories \(P = \{\text{Electronics}, \text{Books}, \text{Apparel}\}\). The "Purchased-From" relation \(R subseteq C imes P\) records transactions:

  • If customer \(c_1\) purchased Electronics and Books, then \((c_1, \text{Electronics}) \in R\) and \((c_1, \text{Books}) \in R\).
  • The Domain represents all active customers who have made at least one purchase.
  • The Range represents all product categories that have generated at least one sale.
  • The Co-Domain is the complete product catalog, which may contain unsought or unsold categories.
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