Number Sets Formulas - Types, Laws & Identities

A set is a well-defined collection of distinct objects or elements. These elements can be numbers, letters, or even real-life objects such as books, fruits, or countries. Sets are usually represented using curly brackets, for example: {1, 2, 3}. In mathematics, sets provide a foundation for organizing data and understanding relationships between groups of numbers.

Basic Operations on Sets

Sets can be combined or compared using operations:

Union: Combines all elements of two sets without repetition.
Intersection: Includes only the common elements between sets.
Difference: Elements present in one set but not in the other.
Complement: Elements not in the given set but present in the universal set.

Identity, Null, and Universal Sets

Identity Set: The set itself when combined with certain operations.
Null Set (Empty Set): A set with no elements, denoted by {}.
Universal Set: The set that contains all possible elements under discussion.

Laws of Sets

Set operations follow specific rules:

Commutative Law: The order of union or intersection does not change the result.
Associative Law: Grouping of sets does not affect the result.
Distributive Law: Intersection distributes over union and vice versa.
De Morgan’s Laws: Show the relationship between union, intersection, and complements.

Real-Life Examples

Union: Combining students who play football and those who play cricket.
Intersection: Students who play both football and cricket.
Difference: Students who play football but not cricket.
Complement: Students outside a particular sports group.

Geometry Formulas | Triangles, Quadrilaterals, Circles & 3D Shapes

1.2 Sets of Numbers

Classification of Numbers

Numbers are grouped into different sets:

Natural Numbers: Counting numbers starting from 1, 2, 3, and so on.
Whole Numbers: Natural numbers along with zero.
Integers: Whole numbers and their negative counterparts.
Rational Numbers: Numbers that can be expressed as a ratio of two integers.
Irrational Numbers: Numbers that cannot be expressed as a ratio, such as √2 or π.
Real Numbers: All rational and irrational numbers.
Imaginary Numbers: Numbers involving the square root of negative values.

Number Line Representation

Numbers can be placed on a number line for visualization. Natural, whole, and integers lie evenly spaced, while rational and irrational numbers fill in the infinite gaps. Imaginary numbers, however, extend beyond the number line into the complex plane.

Comparison and Ordering of Numbers

Numbers can be compared based on their position on the number line. Smaller numbers lie to the left and larger numbers to the right. This ordering helps in solving inequalities and practical calculations.

Real-World Applications

Natural numbers: Counting objects.
Integers: Representing profit and loss.
Rational numbers: Splitting quantities like half or one-third.
Irrational numbers: Engineering designs and circles.
Imaginary numbers: Electrical circuits and wave analysis.

1.3 Basic Identities

Key Arithmetic Identities

Identities are equations that hold true for all values of variables. They are essential tools in algebra and arithmetic for simplification.

Properties of Operations

Addition: Adding zero does not change the number.
Subtraction: Subtracting a number from itself results in zero.
Multiplication: Multiplying by one keeps the number unchanged.
Division: Dividing a number by itself results in one (except zero).

Squares and Cubes

Numbers show patterns in their squares and cubes, such as perfect squares (4, 9, 16) and perfect cubes (8, 27). Recognizing these helps in mental calculations.

Equality and Equivalence

Equality shows that two expressions are the same, while equivalence emphasizes logical similarity in expressions.

Verbal Examples

“The sum of two odd numbers is always even.”
“Multiplying two negative numbers results in a positive number.”

Algebra Formulas | Factoring, Products, Powers & Applications

1.4 Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the idea of real numbers to include imaginary numbers. A complex number has two parts: a real part and an imaginary part.

Real and Imaginary Parts

Real Part: The ordinary number we use daily.
Imaginary Part: Involves the square root of a negative number.

Representation on Argand Plane

The Argand plane is a graphical representation of complex numbers. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. Each complex number can be plotted as a point.

Operations on Complex Numbers

Addition and Subtraction: Combine respective real and imaginary parts.
Multiplication and Division: Follow special rules but remain consistent in structure.