JEE Maths›Sets, Relations & Functions›Question Types

Sets, Relations & Functions - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Set Operations and Cardinality

Pattern

Find n(A union B), n(A cap B), or simplify set expressions using De Morgan's laws

How to recognize

Asks for the number of elements in a union/intersection, or to simplify/verify set identities. May involve Venn diagrams implicitly.

→ Use n(A union B) = n(A) + n(B) - n(A cap B). For three sets, use the full inclusion-exclusion formula. For set identities, apply De Morgan's laws and distributive laws systematically.

Venn Diagram Problems

Pattern

Use Venn diagrams to find the number of elements in specific regions

How to recognize

Mentions a survey, group membership, or asks how many elements are in exactly one set, at least two sets, etc.

→ Draw a Venn diagram and label each disjoint region. Start from the innermost region (intersection of all sets) and work outwards. The sum of all regions equals n(U).

Classify Relations

Pattern

Determine whether a given relation is reflexive, symmetric, transitive, or an equivalence relation

How to recognize

Gives a relation on a set (as a rule or a set of ordered pairs) and asks which properties it satisfies.

→ Check each property independently. For reflexive: verify (a, a) for every element. For symmetric: if (a, b) is in R, check (b, a). For transitive: find all chains (a, b), (b, c) and verify (a, c). An equivalence relation must satisfy all three.

Find Equivalence Classes

Pattern

Find all equivalence classes for a given equivalence relation

How to recognize

Asks for the equivalence class of an element, or the partition induced by an equivalence relation.

→ For each element a, find all elements x such that (a, x) is in R. Group elements that are related to each other. The equivalence classes form a partition of the set (disjoint and exhaustive).

Find Domain and Range

Pattern

Find the domain and/or range of a given function

How to recognize

Asks for the set of valid inputs (domain) or possible outputs (range) of a function involving square roots, logarithms, fractions, or trigonometric expressions.

→ For domain: ensure denominators are nonzero, expressions under square roots are nonneg, log arguments are positive. For range: express x in terms of y and find valid y values, or use the graph/monotonicity.

Check Injective/Surjective

Pattern

Determine if a function is one-one, onto, or bijective

How to recognize

Asks whether a function is injective, surjective, or bijective. May ask to find values of parameters for which a function becomes one-one or onto.

→ One-one: show f(a) = f(b) implies a = b, or show f is strictly monotonic. Onto: show range equals codomain. For polynomial functions, analyze the derivative to check monotonicity.

Composition of Functions

Pattern

Find f(g(x)) or g(f(x)) and determine its domain or properties

How to recognize

Asks for the composite function fog or gof, its domain, or asks to verify properties like associativity or find f given fog and g.

→ Substitute g(x) into f to get fog(x). The domain of fog is all x in dom(g) such that g(x) is in dom(f). To find f from fog and g, let g(x) = t and solve.

Inverse Functions

Pattern

Find the inverse of a function or verify its existence

How to recognize

Asks to find f inverse, or verify that a function is invertible, or asks for (f inverse)(a) for a specific value.

→ First verify f is bijective. To find f inverse: write y = f(x), solve for x in terms of y, then replace y with x. Verify by checking f(f inverse(x)) = x.

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Inverse Trigonometric Functions

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