Sets, Relations, and Functions

1. Core Concept

Sets are the foundation of mathematical language. Relations describe how elements of sets interact, and Functions are specific types of relations where every input has a unique output. IAT often tests the properties of equivalence relations and the invertibility of functions.

2. Key Formulas & Definitions

Subsets: If |A| = n, total subsets = 2ⁿ.
Cartesian Product: A × B = {(a,b) : a∈A, b∈B}. |A × B| = |A| · |B|.
De Morgan’s Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
Cardinality Rule: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
Symmetric Difference: A Δ B = (A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B).

Types of Sets

Finite Set: Limited number of elements (e.g., {1, 2, 3}).
Infinite Set: Unlimited elements (e.g., Set of Natural Numbers ℕ).
Empty Set (∅): A set with no elements. { } or ∅.
Universal Set: Set containing all elements under consideration.
Power Set: Set of all possible subsets. P(A) contains 2ⁿ elements.

3. Types of Relations

A relation R on Set A is:

Reflexive: (a, a) ∈ R for all a ∈ A.
Symmetric: If (a, b) ∈ R, then (b, a) ∈ R.
Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Equivalence Relation: A relation that is Reflexive, Symmetric, and Transitive simultaneously.

Equivalence Classes

If R is an equivalence relation on set A, then all elements related to 'a' form an equivalence class:

[a] = {x ∈ A : (x, a) ∈ R}

These classes partition the set A into disjoint subsets whose union is A.

4. Types of Functions

Injective (One-to-one): Distinct inputs → Distinct outputs. (f(x₁) = f(x₂) &implies; x₁ = x₂).
Surjective (Onto): Range = Co-domain. (Every y in co-domain has an x in domain).
Bijective: Both Injective and Surjective. Inverse f^-1 exists ONLY for Bijective functions.

Domain, Co-domain, and Range

Domain: The set of all valid input values (x).
Co-domain: The set of all possible potential output values (target set).
Range: The set of actual output values produced by the function (f(x)).

Note: Range is always a subset of Co-domain.

(g ∘ f)(x) = g(f(x))

Composition: Generally, g ∘ f ≠ f ∘ g.

Invertibility Condition

A function is invertible if and only if it is Bijective (One-to-one and Onto).

Vertical Line Test: Used to check if a relation is a function.
Horizontal Line Test: Used to check if a function is one-to-one (injective).

5. Special Real-Valued Functions

Modulus: f(x) = |x|. (V-shaped graph, Range: [0, ∞)).
Signum: f(x) = { 1 (x > 0); 0 (x = 0); -1 (x < 0) } (Also expressed as f(x) = x / |x| for x ≠ 0).
Greatest Integer (GIF): f(x) = [x]. (Step function, value is integer ≤ x).

6. Conceptual Insights

Vertical Line Test: A curve is a function if any vertical line crosses it at most once.
Horizontal Line Test: A function is one-to-one if any horizontal line crosses it at most once.
Symmetry of Inverse: The graph of f^-1(x) is the reflection of f(x) about the line y = x.

7. Common Mistakes

Null Set Trap: ∅ is a subset of every set, but {∅} is a set containing the null set (Cardinality = 1).
Transitivity Confusion: A relation is transitive by default if there are no pairs (a, b) and (b, c) to check. (e.g., R={(1,2)} on {1,2,3} is transitive!).
Composition Domain: For (g ∘ f) to exist, the Range of f must be a subset of the Domain of g.

8. IAT Exam Focus Points

Essential Topics:

Equivalence Relations: Practice identifying if a relation (like "divides" or "is parallel to") satisfies R, S, and T.
Domain & Range: Find domains of functions with square roots and denominators (e.g., √(x²-5x+6)).
Composite & Inverse: Be able to calculate f(g(x)) and f^-1(x) for linear/rational functions.
Venn Diagrams: Solve word problems involving 3 sets using the Principle of Inclusion-Exclusion.

9. Practice Mock Test

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