Sets

🎯 Key Points

• Injective (one-one): different inputs → different outputs; Surjective (onto): Range = Codomain; Bijective: both — only bijective functions have an inverse
• De Morgan's laws: (A∪B)' = A'∩B' and (A∩B)' = A'∪B'; inclusion-exclusion: |A∪B| = |A|+|B|-|A∩B|
• |P(A)| = 2ⁿ for a set with n elements (power set = set of ALL subsets, including the empty set and the set itself)
• fog ≠ gof in general (composition is NOT commutative) — always verify by direct substitution, never assume order doesn't matter

Sets & Functions

A set is a well-defined collection of distinct objects called elements. Sets and functions form the foundation of all modern mathematics.

Types of Sets

• Empty set (Null set): Set with no elements: denoted {} or ∅
• Finite set: Has a countable number of elements
• Infinite set: Uncountably or infinitely many elements (e.g., N, R)
• Subset: A ⊆ B if every element of A is in B
• Power set P(A): Set of all subsets of A; |P(A)| = 2^n if |A| = n
• Universal set U: The master set containing all elements under discussion

Set Operations

• Union A ∪ B: Elements in A or B or both
• Intersection A ∩ B: Elements in both A and B
• Complement A': Elements in U but not in A
• Difference A − B: Elements in A but not in B
• Symmetric difference A Δ B: (A − B) ∪ (B − A)

Important Laws

• De Morgan: (A ∪ B)' = A' ∩ B'; (A ∩ B)' = A' ∪ B'
• |A ∪ B| = |A| + |B| − |A ∩ B| (inclusion-exclusion)

Relations and Functions

• A relation from A to B is a subset of A × B (Cartesian product)
• A function f: A → B assigns exactly one element of B to each element of A
• Domain: The set of all valid inputs
• Codomain: The declared output set B
• Range: The actual set of output values (Range ⊆ Codomain)

Types of Functions

• Injective (one-one): Different inputs give different outputs
• Surjective (onto): Every element of codomain has a pre-image
• Bijective: Both injective and surjective
• Composition (fog)(x): f(g(x)): apply g first, then f
• Inverse function f⁻¹: Exists only when f is bijective; f(f⁻¹(x)) = x

Identifying Types of Functions

• One-one (injective): Algebraically, f(x1) = f(x2) must imply x1 = x2. Graphically, every horizontal line cuts the graph at most once.
• Onto (surjective): For every y in the codomain, there must exist some x in the domain with f(x) = y, i.e. Range = Codomain.
• Worked example: Let f: R → R, f(x) = x^2. This is not one-one since f(-2) = f(2) = 4 (two different inputs give the same output), and it is not onto since negative numbers in the codomain (like -1) have no real pre-image. It is also not bijective.
• Worked example: Let f: R → R, f(x) = 2x + 3. For one-one: if 2x1+3 = 2x2+3 then x1 = x2, so it is one-one. For onto: given any y, x = (y-3)/2 always exists in R, so it is onto. Hence f is bijective.

Composition of Functions

Given f(x) = 2x + 3 and g(x) = x^2, the composition (fog)(x) = f(g(x)) = 2x^2 + 3, while (gof)(x) = g(f(x)) = (2x+3)^2 = 4x^2 + 12x + 9.

At x = 3: (fog)(3) = 2(9) + 3 = 21, and (gof)(3) = (9)^2 = 81. Since these differ, composition of functions is not commutative in general, i.e. fog ≠ gof.

Inverse of a Function

• A function f has an inverse f^-1 if and only if f is bijective (both one-one and onto).
• To find the inverse: write y = f(x), solve for x in terms of y, then swap x and y.
• Worked example: For f(x) = 2x + 3, set y = 2x+3, so x = (y-3)/2. Hence f^-1(x) = (x-3)/2. Check: f(5) = 13, and f^-1(13) = (13-3)/2 = 5, confirming the inverse undoes f.
• The graph of f^-1 is the mirror image of the graph of f in the line y = x.

Finding Domain and Range of Common Functions

• Rational functions like f(x) = 1/(x-a): domain is R except x = a (denominator cannot be zero).
• Square root functions like f(x) = sqrt(x-a): domain requires x - a ≥ 0, i.e. x ≥ a; range is [0, ∞).
• Logarithmic functions like f(x) = log(x-a): domain requires x - a > 0, i.e. x > a.
• Quadratic functions like f(x) = x^2: domain is all of R, but range is [0, ∞) since a square can never be negative.
• For a combination of restrictions (e.g. f(x) = sqrt(x-2) + 1/(x-5)), find the domain of each piece separately and take the intersection, also excluding any point that makes a denominator zero.

🚀 JEE Advanced Edge

Restricting the domain to force invertibility: f(x)=x² is not bijective over all of R (fails one-one), but restricting the domain to [0,∞) makes it strictly increasing, hence one-one, and matching the codomain to [0,∞) makes it onto too — this "restrict the domain" technique is exactly how √x is defined as the inverse of x² on the non-negative reals.

Functional equations: A function satisfying f(x+y) = f(x)+f(y) for all real x,y (Cauchy's functional equation) forces f(x) = kx for some constant k, PROVIDED some regularity condition like continuity is assumed — without that condition, pathological non-linear solutions exist, which is why JEE problems on functional equations always include an extra condition (continuity, monotonicity, or a boundary value) to pin down the unique solution.

Worked problem: If f(x) = (x-1)/(x+1), find f(f(x)) and identify what this tells you about f. Approach: f(f(x)) = [(x-1)/(x+1) - 1] / [(x-1)/(x+1) + 1] = [(x-1-x-1)/(x+1)] / [(x-1+x+1)/(x+1)] = (-2)/(2x) = -1/x. Since f(f(x)) ≠ x, f is not self-inverse (f ≠ f⁻¹) — a useful check before assuming a function inverts itself.

Worked Example: Domain and Range

Find the domain and range of f(x) = √(4 − x²).

Domain: 4 − x² ≥ 0 → x² ≤ 4 → domain: [−2, 2]. Range: as x varies over [−2, 2], the value of 4−x² goes from 0 (at x = ±2) to 4 (at x = 0), so f(x) ∈ [0, 2]. Range: [0, 2]. This is the upper semicircle of radius 2.

Worked Example: Composition of Functions

If f(x) = 2x + 1 and g(x) = x², find (f ∘ g)(x) and (g ∘ f)(x).

(f ∘ g)(x) = f(x²) = 2x² + 1.   (g ∘ f)(x) = (2x+1)² = 4x² + 4x + 1. These differ, confirming that f ∘ g ≠ g ∘ f in general — order always matters in composition.

Types of Sets in Detail

• Equal sets: A = B if they have exactly the same elements (order and repetition do not matter): {1,2,3} = {3,1,2}.
• Equivalent sets: have the same number of elements (same cardinality) but not necessarily the same elements.
• Singleton set: a set with exactly one element, e.g. {5}.
• Disjoint sets: A and B with A ∩ B = ∅ (no common element).
• Proper subset: A ⊂ B means A ⊆ B but A ≠ B; every set is a subset of itself but NOT a proper subset of itself.
• The empty set ∅ is a subset of EVERY set; a set with n elements has 2ⁿ subsets and (2ⁿ − 1) proper subsets.

Intervals as Sets

• Open interval (a, b): {x ∈ R : a < x < b} — endpoints excluded.
• Closed interval [a, b]: {x ∈ R : a ≤ x ≤ b} — endpoints included.
• Half-open: [a, b) = {x : a ≤ x < b} and (a, b] = {x : a < x ≤ b}.
• Length of the interval = b − a; infinite intervals: (a, ∞), (−∞, b], (−∞, ∞) = R.
• Intervals are simply subsets of R and can be combined with ∪ and ∩, e.g. [2, 5) ∪ (5, ∞).

Cardinality: Counting Formulas

• Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C).
• n(A only) = n(A) − n(A∩B); n(A − B) = n(A) − n(A ∩ B).
• n(U) = n(A) + n(A′), so n(A′) = n(U) − n(A).
• Worked example: In a class of 40, 25 like tea and 20 like coffee, 10 like both. Number liking at least one = 25 + 20 − 10 = 35, so 40 − 35 = 5 like neither.

Types of Relations

A relation R on a set A (R ⊆ A × A) can have these properties:

• Reflexive: (a, a) ∈ R for every a ∈ A.
• Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R.
• Transitive: (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
• Equivalence relation: reflexive AND symmetric AND transitive (e.g. "is equal to", "is parallel to", "has the same remainder mod n").
• An equivalence relation partitions the set into disjoint equivalence classes.