Sets Relations and Functions formulas for JEE Main & Advanced

22 formula entries for Sets Relations and Functions, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Subset and Superset

A \subseteq B ⟺ \forall x \in A, x \in B

When to use. When comparing sets or proving set relationships.

Trap. Thinking $\subset$ and $\subseteq$ are same— $\subset$ means proper subset (A ≠ B).

Power Set

∣ P (A) ∣ = 2^{∣ A ∣}

When to use. When counting subsets or dealing with all possible combinations of elements.

Trap. Forgetting that empty set and A itself are always in P(A).

Union and Intersection

A \cup B = {x : x \in A or x \in B}

A \cap B = {x : x \in A and x \in B}

When to use. When combining or finding common elements of sets.

Trap. Using 'or' in union as exclusive—it is inclusive.

Difference and Complement

A - B = {x : x \in A and x \notin B}

A^{'} = U - A

When to use. When finding elements exclusive to one set or relative to a universe.

Trap. Thinking $A - B = B - A$ —they are different unless A = B.

De Morgan's Laws

(A \cup B)^{'} = A^{'} \cap B^{'}

(A \cap B)^{'} = A^{'} \cup B^{'}

When to use. When simplifying expressions with complements, unions, and intersections.

Trap. Applying laws incorrectly—e.g., thinking $(A \cup B)^{'} = A^{'} \cup B^{'}$ .

Cardinality of Union (Inclusion-Exclusion)

∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣

∣ A \cup B \cup C ∣ = ∣ A ∣ + ∣ B ∣ + ∣ C ∣ - ∣ A \cap B ∣ - ∣ B \cap C ∣ - ∣ C \cap A ∣ + ∣ A \cap B \cap C ∣

When to use. When counting elements in unions, especially in word problems with overlapping categories.

Trap. Forgetting to subtract intersection in two-set case.

Ordered Pair and Cartesian Product

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

∣ A \times B ∣ = ∣ A ∣ \cdot ∣ B ∣

When to use. When defining relations or functions, or in coordinate geometry.

Trap. Thinking (a,b) = (b,a)—order matters unless a=b.

Relation as a Subset of Cartesian Product

R \subseteq A \times B

When to use. When defining connections between elements of two sets.

Trap. Thinking every subset of $A \times B$ is a function—it's a relation, not necessarily a function.

Types of Relations: Reflexive, Symmetric, Transitive

Reflexive:

\forall a \in A, a R a

; Symmetric:

a R b ⟹ b R a

; Transitive:

a R b \land b R c ⟹ a R c

When to use. When classifying relations as equivalence or partial order.

Trap. Assuming a relation is reflexive if it has some self-pairs—it must have all.

Equivalence Relation and Equivalence Class

Equivalence class of a:

[a] = {x \in A : x R a}

When to use. When grouping elements that share a property, e.g., congruence modulo n.

Trap. Thinking an equivalence relation must be antisymmetric—it's not.

Function as a Special Relation

f \subseteq A \times B

such that

\forall a \in A, \exists! b \in B

with

(a, b) \in f

When to use. When defining a mapping from one set to another.

Trap. Thinking a function can have one input with multiple outputs—it cannot.

Domain, Codomain, and Range of a Function

Domain =

{x : f (x) defined}

, Range =

{f (x) : x \in Domain}

When to use. When analyzing a function's behavior or solving equations.

Trap. Thinking range equals codomain—range is subset, not necessarily equal.

One-One (Injective) Function

f (a) = f (b) ⟹ a = b

When to use. When determining if a function has an inverse or is invertible.

Trap. Judging one-one-ness without fixing the domain; e.g. $x^{2}$ is one-one on $[0, \infty)$ but not on $R$ .

Onto (Surjective) Function

\forall y \in B, \exists x \in A

such that

f (x) = y

When to use. When checking if a function covers all possible outputs.

Trap. Thinking onto means every element of domain maps to something—that's always true for functions.

Bijective Function and Inverse

f^{- 1} (y) = x

f (x) = y

, for bijective

f

When to use. When finding inverse functions or solving functional equations.

Trap. Assuming any function has an inverse—only bijective ones do.

Composition of Functions

(g \circ f) (x) = g (f (x))

When to use. When combining functions or solving functional equations.

Trap. Thinking composition is commutative— $f \circ g \neq g \circ f$ generally.

Finding Domain of a Function

For

f (x) = \frac{1}{g (x)}

g (x) \neq 0

; for

\sqrt{g (x)}

g (x) \geq 0

; for

\log (g (x))

g (x) > 0

When to use. When given a function expression and asked for its natural domain.

Trap. Forgetting to consider both numerator and denominator in rational functions.

Finding Range of a Function

For quadratic

f (x) = a x^{2} + b x + c

, range is

[f (- b / 2 a), \infty)

if a>0, else

(- \infty, f (- b / 2 a)]

When to use. When asked to find set of all possible outputs of a function.

Trap. Assuming range is all reals for rational functions—check for horizontal asymptotes.

Checking Injectivity and Surjectivity

f^{'} (x) > 0

(or

< 0

) throughout an interval domain, then

f

is strictly monotonic, hence injective.

When to use. When classifying functions or proving bijectivity.

Trap. Strictly monotonic implies injective with no continuity needed, but the converse is false (an injective function need not be monotonic); the derivative test needs an interval domain where $f^{'}$ keeps one sign.

Solving Functional Equations

For Cauchy equation

f (x + y) = f (x) + f (y)

, if continuous,

f (x) = c x

When to use. When given a functional equation and asked to find f(x) or its properties.

Trap. Assuming all functional equations have simple polynomial solutions—some require continuity.

Standard Sets and Notations

N \subset Z \subset Q \subset R \subset C

When to use. When working with any set theory or function problems.

Trap. Some texts include 0 in N—check context; JEE usually uses N = {1,2,3,...}.

Venn Diagrams for Set Problems

For two sets, draw two overlapping circles; for three, three overlapping circles.

When to use. When solving problems involving multiple set relationships, especially with given cardinalities.

Trap. Forgetting to label regions correctly—each region represents a unique combination.

Read the full chapter: Sets Relations and Functions for JEE, with what JEE asks and the common traps.