Relations and Functions

Relations and functions are fundamental concepts in mathematics that describe connections between elements of sets. A relation from set \(A\) to set \(B\) is a subset of the Cartesian product \(A \times B\), consisting of ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\). A function is a special type of relation where each element of \(A\) is related to exactly one element of \(B\).

Concept Explanation

Given two non-empty sets \(A\) and \(B\), the Cartesian product \(A \times B\) is defined as:

\[ A \times B = \{ (a, b) : a \in A, b \in B \} \]

This set contains all ordered pairs where the first element is from \(A\) and the second from \(B\). A relation \(R\) from \(A\) to \(B\) is any subset of \(A \times B\).

The domain of a relation \(R\) is the set of all first elements of the ordered pairs in \(R\), and the range is the set of all second elements:

\[ \text{Domain}(R) = \{ a : (a, b) \in R \} \quad \text{and} \quad \text{Range}(R) = \{ b : (a, b) \in R \} \]

The codomain is the set \(B\) itself.

Types of Relations

• Void Relation: The empty set \(\emptyset\) is a relation with no elements.
• Universal Relation: The entire Cartesian product \(A \times B\).
• Identity Relation: \(I_A = \{ (a, a) : a \in A \} \).
• Reflexive Relation: Every element relates to itself: \((a, a) \in R, \forall a \in A\).
• Symmetric Relation: If \((a, b) \in R\), then \((b, a) \in R\).
• Anti-Symmetric Relation: If \((a, b) \in R\) and \((b, a) \in R\), then \(a = b\).
• Transitive Relation: If \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\).
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
• Partial Order Relation: Reflexive, anti-symmetric, and transitive.
• Total Order Relation: A partial order relation where every pair of elements is comparable.

Worked Illustration

Let \(A = \{1, 2, 3\}\) and \(B = \{f, g\}\). The Cartesian product \(A \times B\) is:

\[ A \times B = \{ (1, f), (1, g), (2, f), (2, g), (3, f), (3, g) \} \]

A relation \(R\) could be \(\{ (1, f), (2, g), (3, f) \}\). The domain of \(R\) is \(\{1, 2, 3\}\), and the range is \(\{f, g\}\).

Solved Example

Example: Determine if the relation \(R = \{(1,1), (2,2), (3,3), (1,2)\}\) on \(A = \{1, 2, 3\}\) is reflexive, symmetric, and transitive.

Solution:

• Reflexive: Check if \((a, a) \in R\) for all \(a \in A\). Here, \((1,1), (2,2), (3,3) \in R\), so \(R\) is reflexive.
• Symmetric: For each \((a, b) \in R\), check if \((b, a) \in R\). \((1,2) \in R\) but \((2,1) \notin R\), so \(R\) is not symmetric.
• Transitive: Check if whenever \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\). \((1,2) \in R\) and \((2,2) \in R\), so \((1,2) \in R\) which is true. No other pairs violate transitivity, so \(R\) is transitive.

Practice Set

Level 1 – Easy

• Find the Cartesian product of \(A = \{x, y\}\) and \(B = \{1, 2\}\).
• Identify the domain and range of \(R = \{(a, 3), (b, 4), (c, 3)\}\).
• Is the relation \(R = \{(1,1), (2,2)\}\) on \(\{1,2\}\) reflexive?

Level 2 – Moderate

• Given \(R = \{(1,2), (2,3), (1,3)\}\) on \(\{1,2,3\}\), check if \(R\) is transitive.
• Determine if \(R = \{(a,a), (b,b), (a,b)\}\) on \(\{a,b\}\) is symmetric.
• Find the number of elements in \(A \times B\) if \(n(A) = 4\) and \(n(B) = 3\).

Level 3 – Challenging

• Prove that the identity relation on a set \(A\) is an equivalence relation.
• Show that the empty relation on \(A\) is symmetric and transitive but not reflexive.
• Given \(R\) is a partial order on \(A\), prove that \(R\) is reflexive, anti-symmetric, and transitive.

Answer Key

• Level 1:
  • \(A \times B = \{(x,1), (x,2), (y,1), (y,2)\}\)
  • Domain = \(\{a, b, c\}\), Range = \(\{3, 4\}\)
  • Yes, \(R\) is reflexive on \(\{1,2\}\)
• Level 2:
  • \(R\) is transitive because \((1,2)\) and \((2,3)\) imply \((1,3)\) which is in \(R\)
  • \(R\) is not symmetric because \((a,b) \in R\) but \((b,a) \notin R\)
  • Number of elements in \(A \times B = 4 \times 3 = 12\)
• Level 3:
  • Identity relation is reflexive, symmetric, and transitive by definition.
  • Empty relation has no elements, so symmetric and transitive hold vacuously, but not reflexive as \((a,a)\) not in \(\emptyset\).
  • By definition, partial order is reflexive, anti-symmetric, and transitive.

Quick Reference

Term	Definition
Cartesian Product	Set of all ordered pairs \((a,b)\) with \(a \in A\), \(b \in B\)
Relation	Subset of \(A \times B\)
Function	Relation with exactly one output for each input
Domain	Set of all first elements in relation
Range	Set of all second elements in relation
Reflexive	\((a,a) \in R\) for all \(a\)
Symmetric	If \((a,b) \in R\), then \((b,a) \in R\)
Transitive	If \((a,b)\) and \((b,c)\) in \(R\), then \((a,c)\) in \(R\)

Glossary

• Ordered Pair: A pair \((a,b)\) where order matters.
• Subset: A set all of whose elements belong to another set.
• Reflexive Relation: Relation where every element relates to itself.
• Symmetric Relation: Relation where if \(a\) relates to \(b\), then \(b\) relates to \(a\).
• Transitive Relation: Relation where if \(a\) relates to \(b\) and \(b\) relates to \(c\), then \(a\) relates to \(c\).
• Equivalence Relation: Relation that is reflexive, symmetric, and transitive.
• Partial Order: Relation that is reflexive, anti-symmetric, and transitive.

Functions and Their Types

A function \(f\) from set \(A\) to set \(B\), denoted \(f : A \to B\), assigns to each element \(a \in A\) exactly one element \(b \in B\), called the image of \(a\), written \(f(a) = b\). The set \(A\) is the domain, \(B\) is the codomain, and the set of all images \(f(a)\) is the range.

Concept Explanation

Functions can be classified based on their mapping properties:

• One-One (Injective): Different elements in \(A\) map to different elements in \(B\).
• Onto (Surjective): Every element in \(B\) is an image of some element in \(A\).
• One-One Onto (Bijective): Both injective and surjective.
• Many-One: Multiple elements in \(A\) map to the same element in \(B\).

Formula Derivation

Number of functions from a set \(A\) with \(p\) elements to a set \(B\) with \(q\) elements is:

\[ q^p \]

since each element of \(A\) can be mapped to any of the \(q\) elements of \(B\).

Worked Illustrations

Identity Function: \(f : \mathbb{R} \to \mathbb{R}\), defined by \(f(x) = x\). Graph is a straight line through origin with slope 1.

Constant Function: \(f(x) = c\), where \(c\) is constant. Graph is a horizontal line at \(y = c\).

Modulus Function: \(f(x) = |x|\), defined as:

\[ f(x) = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases} \]

Graph is V-shaped with vertex at origin.

Signum Function: \(f(x) = \frac{|x|}{x}\) for \(x \neq 0\), and \(f(0) = 0\). It takes values \(-1, 0, 1\) depending on sign of \(x\).

Greatest Integer Function: \(f(x) = [x]\), the greatest integer less than or equal to \(x\). Graph is a step function.

Polynomial Functions: Examples include \(f(x) = x^2\) (parabola) and \(f(x) = x^3\) (cubic curve).

Exponential Function: \(f(x) = a^x\), where \(a > 0, a \neq 1\). Domain is \(\mathbb{R}\), range is \((0, \infty)\).

Natural Exponential Function: \(f(x) = e^x\), where \(e \approx 2.718\).

Logarithmic Function: \(f(x) = \log_a x\), inverse of exponential function, with domain \((0, \infty)\) and range \(\mathbb{R}\).

Solved Examples

Example 1: Find the domain and range of \(f(x) = |x|\).

Solution: Domain is all real numbers \(\mathbb{R}\). Range is \([0, \infty)\) since absolute value is always non-negative.

Example 2: Determine if \(f(x) = x^3\) is one-one and onto from \(\mathbb{R}\) to \(\mathbb{R}\).

Solution: \(f(x) = x^3\) is strictly increasing, so it is one-one. For any real \(y\), \(x = \sqrt[3]{y}\) satisfies \(f(x) = y\), so it is onto. Hence, \(f\) is bijective.

Practice Set

Level 1 – Easy

• Find the range of \(f(x) = 5\) for all real \(x\).
• Evaluate \(f(-3)\) for \(f(x) = |x|\).
• Is \(f(x) = x^2\) one-one on \(\mathbb{R}\)?

Level 2 – Moderate

• Find the domain and range of \(f(x) = \log_2 x\).
• Show that \(f(x) = e^x\) is one-one and onto \((0, \infty)\).
• Find the number of functions from a set with 3 elements to a set with 2 elements.

Level 3 – Challenging

• Prove that the composition of two one-one functions is one-one.
• Show that the inverse of a bijective function is also a function.
• Prove that the sum of two even functions is even.

Answer Key

• Level 1:
  • Range of \(f(x) = 5\) is \(\{5\}\).
  • \(f(-3) = |-3| = 3\).
  • \(f(x) = x^2\) is not one-one on \(\mathbb{R}\) because \(f(2) = f(-2) = 4\).
• Level 2:
  • Domain of \(\log_2 x\) is \((0, \infty)\), range is \(\mathbb{R}\).
  • \(f(x) = e^x\) is one-one because it is strictly increasing; onto \((0, \infty)\) because \(e^x > 0\) for all \(x\).
  • Number of functions = \(2^3 = 8\).
• Level 3:
  • Composition of one-one functions is one-one by definition.
  • Inverse of bijection is well-defined and a function.
  • Sum of even functions \(f\) and \(g\) satisfies \((f+g)(-x) = f(-x) + g(-x) = f(x) + g(x) = (f+g)(x)\).

Quick Reference

Function Type	Definition	Domain	Range
Identity	\(f(x) = x\)	\(\mathbb{R}\)	\(\mathbb{R}\)
Constant	\(f(x) = c\)	\(\mathbb{R}\)	\{c\}
Modulus	\(f(x) = |x|\)	\(\mathbb{R}\)	\([0, \infty)\)
Signum	\(f(x) = \frac{|x|}{x}\)	\(\mathbb{R}\setminus\{0\}\)	\{-1, 1\}
Greatest Integer	\(f(x) = [x]\)	\(\mathbb{R}\)	\(\mathbb{Z}\)
Exponential	\(f(x) = a^x, a>0, a \neq 1\)	\(\mathbb{R}\)	\((0, \infty)\)
Logarithmic	\(f(x) = \log_a x, a>0, a \neq 1\)	\((0, \infty)\)	\(\mathbb{R}\)

Glossary

• Domain: Set of inputs for a function.
• Range: Set of outputs of a function.
• Codomain: Target set of a function.
• One-One (Injective): Different inputs map to different outputs.
• Onto (Surjective): Every element of codomain is an output.
• Bijective: Both one-one and onto.
• Composition: Applying one function after another.
• Even Function: \(f(-x) = f(x)\).
• Odd Function: \(f(-x) = -f(x)\).