Relations and Functions revision notes

Relations

Functions

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 Tip If 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 Tip Transitivity 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.

• 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.

• 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 Tip This 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 ($gof$): 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 Tip To prove a function is invertible without finding its inverse, simply prove it is both one-one and onto.

• 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.

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.

• 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 $f$.

• The Vacuous Truth of Transitivity: A relation where we have $(1,2)$ but no pairs starting with $2$ is still transitive. You cannot say transitivity fails unless you explicitly find a broken chain (i.e., you have $(a,b)$ and $(b,c)$ but lack $(a,c)$).
• Intersection of Equivalence Relations: If $R_1$ and $R_2$ are equivalence relations in a set $A$, then their intersection $R_1 \cap R_2$ is ALWAYS an equivalence relation. JEE Tip However, their union $R_1 \cup R_2$ is NOT necessarily an equivalence relation (it may fail transitivity).
• Function Composition Domain Traps: $gof(x)$ is only defined if the range of $f(x)$ is a subset of the domain of $g(x)$. Blindly substituting algebraic formulas without checking this restriction leads to wrong domain answers.
• Non-Commutative Composition: $gof$ and $fog$ are generally not equal ($gof \neq fog$). For instance, if $f(x) = \cos x$ and $g(x) = 3x^2$, $gof(x) = 3\cos^2x$ while $fog(x) = \cos(3x^2)$.

• 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).