Relations and Functions revision notes

Key Concepts & Definitions

Relations

Relation:

A mathematical connection or link between two objects. A relation RRR from a set AAA to a set BBB is defined mathematically as an arbitrary subset of the Cartesian product A×BA \times BA×B. If (a,b)∈R(a, b) \in R(a,b)∈R, we say that aaa is related to bbb under the relation RRR, written as aRba R baRb.

Empty Relation:

A relation RRR in a set AAA where no element of AAA is related to any element of AAA, i.e., R=ϕ⊂A×AR = \phi \subset A \times AR=ϕ⊂A×A.

Universal Relation:

A relation RRR in a set AAA where each element of AAA is related to every element of AAA, i.e., R=A×AR = A \times AR=A×A.

Trivial Relations:

Both the empty relation and the universal relation are sometimes called trivial relations.

Identity Relation:

A relation where every element is related only to itself. I={(a,a):a∈A}I = \{(a,a) : a \in A\}I={(a,a):a∈A}.

Functions

• Function: Functions are a special kind of relation. The concept evolved historically: Descartes first used the term to denote powers of $x$; Leibnitz introduced the phrase "function of $x$"; Euler introduced the standard notation $f(x)$, $F(x)$, etc.; and Dirichlet provided the rigorous foundation for the modern set-theoretic definition abstraction.
• Domain, Co-domain, and Range: The set of all valid inputs is the domain, the set in which the outputs fall is the co-domain, and the set of actual outputs is the range.
• Equal Functions: Two functions $f: A \to B$ and $g: A \to B$ are called equal functions if their domains and codomains are identical, and $f(a) = g(a)$ for all $a \in A$.

Types of Relations

To study equivalence, we classify relations in a set $A$ into three specific types:

• Reflexive Relation: $(a, a) \in R$ for every $a \in A$.JEE TIPIf even a single element $a \in A$ does not have $(a,a) \in R$, the relation is NOT reflexive.
• Symmetric Relation: $(a, b) \in R$ implies that $(b, a) \in R$ for all $a, b \in A$.
• Transitive Relation: $(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$ for all $a, b, c \in A$.JEE TIPTransitivity ONLY fails if $(a,b)$ and $(b,c)$ exist, but $(a,c)$ does NOT. If $(a,b)$ exists but there is no $(b,c)$ pairing to check, the relation is vacuously transitive!

Equivalence Relation & Equivalence Classes

• Equivalence Relation: A relation $R$ in a set $A$ is an equivalence relation if it is reflexive, symmetric, and transitive.
• Equivalence Classes (Partitions): Given an arbitrary equivalence relation $R$ in an arbitrary set $X$, $R$ divides $X$ into mutually disjoint subsets $A_i$ called partitions or subdivisions.
  • All elements of $A_i$ are related to each other.
  • No element of $A_i$ is related to any element of $A_j$ ($i \neq j$).
  • The union of all $A_i$ is $X$, and their intersection is empty ($\phi$).
  • The subset $A_i$ containing an element $a$ is called an equivalence class and is denoted by $[a]$.

Types of Functions

• One-One (Injective) Function: A function $f: X \to Y$ is one-one if the images of distinct elements of $X$ are distinct. Mathematically: for every $x_1, x_2 \in X$, $f(x_1) = f(x_2)$ implies $x_1 = x_2$. If it is not one-one, it is called many-one.
• Onto (Surjective) Function: A function $f: X \to Y$ is onto if every element of $Y$ is the image of some element of $X$ under $f$. This means for every $y \in Y$, there exists an $x \in X$ such that $f(x) = y$.
  • Rule: $f: X \to Y$ is onto if and only if Range of $f = Y$.
• Bijective Function: A function $f: X \to Y$ is bijective if it is both one-one and onto.

Finite vs. Infinite Sets Property

• For an arbitrary finite set $X$, a one-one function $f: X \to X$ is necessarily onto, and an onto map $f: X \to X$ is necessarily one-one.
• JEE TIPThis is a characteristic difference between a finite and an infinite set. For an infinite set (like $N$ or $R$), a function can be one-one but not onto (e.g., $f(x) = 2x$ on $N$), or onto but not one-one.

Composition of Functions & Invertibility

• Composition of Functions ($gof$): Let $f: A \to B$ and $g: B \to C$ be two functions. The composition of $f$ and $g$, denoted by $gof$, is the function $gof: A \to C$ given by $gof(x) = g(f(x)), \forall x \in A$.JEE TIP$gof \neq fog$ in general. Matrix multiplication and function composition are highly analogous (both are associative but non-commutative).
• Invertible Function: A function $f: X \to Y$ is invertible if there exists a function $g: Y \to X$ such that $gof = I_X$ and $fog = I_Y$. Here, $I_X$ and $I_Y$ are identity functions.
• Inverse: The function $g$ is called the inverse of $f$ and is denoted by $f^{-1}$.
• Condition for Invertibility: If $f$ is invertible, then $f$ must be one-one and onto (bijective). Conversely, if $f$ is one-one and onto, then $f$ must be invertible.JEE TIPTo prove a function is invertible without finding its inverse, simply prove it is both one-one and onto.

Advanced Function Properties

• Even and Odd Functions:
  • Even: $f(-x) = f(x)$. Graph is symmetric about the y-axis.
  • Odd: $f(-x) = -f(x)$. Graph is symmetric about the origin.
  • Every function can be uniquely expressed as the sum of an even and an odd function: $f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$.
• Periodic Functions: A function $f(x)$ is periodic if there exists a positive real number $T$ such that $f(x+T) = f(x)$ for all $x$ in the domain. The smallest such $T$ is the fundamental period.

Combinatorics of Relations & Functions

Let set $A$ have $m$ elements and set $B$ have $n$ elements.

• Total number of relations from $A$ to $B$ is $2^{mn}$.
• Total number of functions from $A$ to $B$ is $n^m$.
• Total number of injective (one-one) functions from $A$ to $B$ is $^nP_m$ (where $n \ge m$). If $n < m$, it is $0$.
• Total number of bijective functions from $A$ to $A$ is $m!$ (this matches the permutation on symbols principle).
• Total number of reflexive relations on set $A$ is $2^{m(m-1)}$.
• Total number of symmetric relations on set $A$ is $2^{m(m+1)/2}$.
• Specific Summations: The number of relations containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but NOT symmetric on set $\{1, 2, 3\}$ is exactly three. The number of equivalence relations containing $(1, 2)$ and $(2, 1)$ on set $\{1, 2, 3\}$ is exactly two.

Conditions & Limitations

• Finite versus Infinite Sets: The theorem "$f: X \to X$ is one-one if and only if it is onto" strictly applies ONLY to finite sets. Do not apply this to sets like $\mathbb{R}, \mathbb{Z}$, or $\mathbb{N}$.
• Addition of Functions vs Co-Domain Limits: Let $I_N$ be the identity function on $\mathbb{N}$. $I_N$ is onto. However, $I_N + I_N$ (which evaluates to $2x$) is NOT onto because odd numbers in the co-domain $\mathbb{N}$ lose their pre-image (e.g. $2x = 3$ has no solution in $\mathbb{N}$).
• Greatest Integer Function limitations: $f(x) = [x]$ on $\mathbb{R} \to \mathbb{R}$ is neither one-one nor onto.
• Modulus & Signum Function limitations: Modulus function $f(x) = \|x\|$ and Signum function $f(x) = \text{sgn}(x)$ defined on $\mathbb{R} \to \mathbb{R}$ are strongly many-one and into (not onto), as they collapse multiple real values to single outputs and have heavily restricted ranges.
• Domain restrictions in Inverse: Whenever solving $f(x) = y$ to find the inverse $g(y)$, you must check if the derived $x$ falls strictly within the defined domain of $f$.

Previous Year JEE Topics

• Checking Injectivity using Calculus: Differentiating $f(x)$ to check if $f'(x) > 0$ or $f'(x) < 0$ continuously across the domain.
• Number of Possible Functions/Relations: Combinatorics problems linking P&C with relations (e.g., number of reflexive and symmetric relations).
• Equivalence Relations: Matrix/Coordinate-based equivalence conditions. Example: $(x, y) R (u, v) \iff xv = yu$.
• Finding the Inverse: Solving algebraically for $x = f^{-1}(y)$, and confirming the correct branch of a quadratic/trigonometric function based on domain boundaries.
• Functional Equations: Guessing the function profile based on relationships like $f(x+y) = f(x)f(y)$ (implying exponential functions).

Memory Aids & JEE Traps

• JEE TIPWhen checking onto functions, students look at the formula and forget to check the declared co-domain. $f(x) = 2x$ is onto if $f: \mathbb{R} \to \mathbb{R}$, but it is NOT onto if $f: \mathbb{N} \to \mathbb{N}$ because there's no pre-image for odd numbers like 1.
• JEE TIP$f^{-1}(x)$ means the inverse function, NOT $\frac{1}{f(x)}$. The inverse undoes the operation; the reciprocal divides it.
• JEE TIPAn even function $f(x) = f(-x)$ maps two inputs to the same output. Therefore, it is many-one, and its global inverse never exists unless the domain is restricted to $x \ge 0$ or $x \le 0$.

Top 10 JEE MCQ Traps (Misconception → Correct Understanding)

1. Misconception: If a relation is not symmetric, it must be anti-symmetric. → Correct Understanding: Symmetry and anti-symmetry are not polar opposites. A relation can easily be neither symmetric nor anti-symmetric (e.g., if it contains $(1,2)$ and $(2,1)$ but also contains $(1,3)$ without $(3,1)$).
2. Misconception: Transitivity fails if we only have the pair $(a,b)$ and nothing else. → Correct Understanding: Transitivity ONLY fails if we have $(a,b)$ AND $(b,c)$ but are missing $(a,c)$. If there is no second pair starting with $b$, the relation is vacuously transitive.
3. Misconception: Any function $f: A \to B$ is onto if the cardinalities $|A| = |B|$. → Correct Understanding: This is only true if $A$ and $B$ are FINITE sets AND the function is one-one. For infinite sets, $|A| = |B|$ does not guarantee an injective function is surjective.
4. Misconception: The empty relation is an equivalence relation on an empty set. → Correct Understanding: The empty relation on a non-empty set is symmetric and transitive (vacuously) but it is NEVER reflexive because $(a,a)$ is missing for elements in the set.
5. Misconception: $gof(x)$ and $fog(x)$ always share the same domain limits. → Correct Understanding: The domain of $gof(x)$ is strictly constrained by the domain of $f(x)$, whereas the domain of $fog(x)$ is constrained by the domain of $g(x)$. They are completely independent.
6. Misconception: If $f(x)$ is an even function, its inverse can be found via algebra. → Correct Understanding: Even functions are globally many-one (since $f(x) = f(-x)$), meaning they are not bijective. Their global inverse DOES NOT exist.
7. Misconception: $f(x) = x^2$ is a strictly non-invertible function. → Correct Understanding: Invertibility strictly depends on the domain and codomain. While $f: \mathbb{R} \to \mathbb{R}$ is neither 1-1 nor onto, $f: [0, \infty) \to [0, \infty)$ is bijective and fully invertible.
8. Misconception: Composition of functions is commutative. → Correct Understanding: Function composition is associative but generally NON-commutative ($fog \neq gof$).
9. Misconception: To find the inverse of $f(x)$, just compute $(f(x))^{-1}$ or $\frac{1}{f(x)}$. → Correct Understanding: The inverse $f^{-1}(x)$ is the function that reverses the mapping (reflecting the graph across the line $y=x$). It is completely different from the algebraic reciprocal.
10. Misconception: A reflexive relation is identical to the identity relation. → Correct Understanding: The identity relation contains only the pairs $(a,a)$. A reflexive relation MUST contain at least all $(a,a)$ pairs, but it is free to contain other cross-pairs $(a,b)$ as well.