सेट्स और सेट संचालन

Sets

• Sets are collections of distinct objects, often numbers.
• Sets are usually denoted by capital letters (e.g., A, B).
• Elements of a set are listed within curly braces { }.
• Example: Set A = {1, 2, 3}

Types of Sets

• Finite Sets: Sets with a limited number of elements. Example: {1, 2, 3}
• Infinite Sets: Sets with an unlimited number of elements. Example: The set of all natural numbers (1, 2, 3, ...)
• Empty Set: A set with no elements, denoted by {} or Ø.
• Universal Set (U): The set containing all elements under consideration in a specific context.

Set Operations

• Union (∪): Combines elements from two or more sets. The union of sets A and B (A ∪ B) contains all elements present in either A or B or both. Example: A = {1, 2}, B = {2, 3}, A ∪ B = {1, 2, 3}
• Intersection (∩): Finds elements common to two or more sets. The intersection of sets A and B (A ∩ B) contains only the elements present in both A and B. Example: A = {1, 2}, B = {2, 3}, A ∩ B = {2}
• Complement ('): Contains elements not in a given set, relative to a universal set. Example: If U = {1, 2, 3, 4, 5} and A = {1, 3}, then A' = {2, 4, 5}

Number Systems

• Natural Numbers (N): Positive whole numbers (1, 2, 3,...).
• Whole Numbers (W): Natural numbers plus zero (0, 1, 2, 3,...).
• Integers (Z): Whole numbers, their opposites and zero (-3, -2, -1, 0, 1, 2, 3,...).
• Rational Numbers (Q): Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Example: 1/2, 3, -2/5
• Irrational Numbers (I): Numbers that cannot be expressed as a fraction of two integers. Example: π (pi), √2.
• Real Numbers (R): The combination of rational and irrational numbers.
• Complex Numbers (C): Numbers that can be written in the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit (√(-1)).

Basic Algebraic Structures

• Fields: A set with two operations (addition and multiplication) satisfying specific properties, like associativity, commutativity, and distributivity. Examples include real numbers and rational numbers.
• Groups: A set with an operation satisfying properties like closure, associativity, the existence of an identity element, and the existence of an inverse for each element.
• Rings: A set with two operations (addition and multiplication) satisfying properties similar to groups, but with the additional property that multiplication distributes over addition and is often not commutative.
• Fields and groups are subsets of rings

Fundamental Principles of Counting

• The Multiplication Principle: If one task can be performed in m ways and a second task can be performed in n ways, then both tasks can be performed in m x n ways.
• Permutations: An arrangement of items in a specific order. Formula: nPr = n! / (n-r)!
• Combinations: A selection of items without regard to order. Formula: nCr = n! / (r! * (n-r)!)

Equations and Inequalities

• Linear Equations: Equations that can be written in the form ax + b = 0.
• Quadratic Equations: Equations that can be written in the form ax² + bx + c = 0.
• Solving inequalities: Finding values that satisfy inequalities (e.g., x > 3, x ≤ 5).

Polynomials

• Polynomials are expressions consisting of variables and coefficients, combined through addition, subtraction, and multiplication.
• Polynomials have integer exponents.
• Polynomials are generally denoted (axⁿ + bx⁽ⁿ⁻¹⁾ + cx⁽ⁿ⁻²⁾…), where the exponent of x is integer.

Geometry

• Points, lines, and planes are the fundamental units of geometry.
• Angles, lengths, and distances are measured in geometric contexts.
• Triangles, quadrilaterals, and circles are frequently studied geometric shapes.

Trigonometry

• Trigonometry studies angles and side lengths of triangles.
• Sine, cosine, and tangent are fundamental trigonometric functions.
• Trigonometry is applicable to many real-world problems.

Calculus

• Limits: The concept of a function approaching a specific value as the input approaches a certain point.
• Derivatives: Measure the instantaneous rate of change of a function.
• Integrals: Find the area under a curve or total value across a range of inputs.

Statistics

• Descriptive Statistics: Uses measures like mean, median, mode, and standard deviation to describe data.
• Inferential Statistics: Draw conclusions about a population based on a sample of data.

Discrete Mathematics

• Logic: Propositions, connectives (and, or, not), truth tables.
• Set Theory: Sets, operations (union, intersection, complement).
• Graph Theory: Graphs, paths, connectivity.
• Counting Principles: Permutations and combinations.
• Recurrence Relations: Methods to describe sequences with recursive definitions.