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Sets Math

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrating Even and Odd Functions
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Differential Equation
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- One-Sided Limits
- Optimization Problems
- P Series
- Particular Solutions to Differential Equations
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal´s Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

A set can contain anything, be it a collection of numbers, days of the week, fruits, etc. A simple example is a set of positive integers up to 5, which looks like this: {1, 2, 3, 4, 5}. But exactly how do we define and use sets? Let's take a look.

Sets in mathematics are an organised collection of objects called elements. They are noted mathematically by curly brackets {}.

Elements of sets can be represented using various notations, the roster, or the set builder, and we will return to this later.

Specific symbols are used to describe given sets. Below are common ones and their meaning.

symbol | Meaning |

U | Universal set |

n (x) | Cardinal number of set X |

{} | Denotes a set |

∈ | Is an element of |

∉ | Is not an element of |

∅ | Empty set or null |

⋃ | union |

∩ | Intersection |

⊆ | Subset |

⊇ | Superset |

| | Such that |

The items contained in a set are called the elements of the set. They are denoted by curly brackets with commas separating each element. We can use specific notation to denote that something is an element of a particular set. For example, if we had A = {1, 2, 3, 4}, we could write that 3 ∈ A, which means 3 is an element of A. However, as it is evident that 5 is not a member of A, that can be denoted as 5 ∉ A.

- Here are examples of commonly used sets.
N - Set of all natural numbers

Z - Set of all integers

Q - Set of all rational numbers

R - Set of all real numbers

Z + - Set of all positive integers

To define a set, it must be a collection of unique elements. A significant property of sets is that the elements somehow should be related to each other or share a common property. For example, by defining a list of primary colours in a set, we mean that all the elements are primary colours.

Cardinality denotes the total number of elements in a set. This means that if we have a set of natural numbers below 6, the cardinality of that set will be 5. Take our set to be A = {1, 2, 3, 4, 5} – there are five elements present in the set. That makes our cardinality 5. The cardinality of A is denoted by | A | or n (A), where n is the number of elements in the brackets, and A is any given set.

There are various ways sets can be represented. The fundamental difference is in the way the elements are listed. They can be represented by either semantic, roster, or set builder form.

This notation is a statement form of describing the elements of a set. For example, we can list natural prime numbers below 20. Another example is the list of the months in a year. In semantic form, they can even be written as {set of odd natural numbers less than 10}.

The roster form is the most commonly used notation used for sets. Elements are denoted by curly brackets and separated by commas. With this type of notation, the elements of the set are usually mentioned. For example, a set of odd natural numbers below 10 will be A = {1, 3, 5, 7, 9}. Equating A to our set means that anywhere we find A, we are talking about our list of odd natural numbers.

In another instance, you have a set with infinite elements, which is usually expressed with a series of dots at the end of the last stated element. For example, a set of positive integers will be denoted by = {1, 2, 3, 4, 5, ....} This means there are endless numbers following 5 in the order that has been already expressed.

This mathematical notation is used to describe sets by demonstrating the properties that its members must satisfy. There is usually a statement that specifically describes the common feature of all the elements of a set.

For example, a set of positive integers up to 5 can be denoted by the set builder as .

Another example could be . This notation states that all the elements of set A are even numbers that are less than or equal to 12. By writing this in roster form, we will have A = {2, 4, 6, 8, 10, 12}, and its cardinality will be 6.

There are many different types of sets in mathematics. We will go over them in this section.

Sets that do not contain any elements are called empty sets or null sets. They are denoted by either {} or ∅.

These types of sets have only one element contained in them. They are also called unit sets. For example, A = {4}

Finite sets are sets with a countable number of elements in them. For example, A = {a set of positive integers below 7} will be A = {1, 2, 3, 4, 5, 6} or {x | x is positive integer <7}.

These are sets that contain an infinite number of elements. An example of this set is Z = {set of all integers}. Another example is multiples of three. Which can be denoted by C = {3, 6,9, 12, 15, .....}. The series of dots after the last element listed is used to express its infinite status.

Two sets are said to be equal when they contain the same elements. The order in which they are arranged does not matter. For example, if I had two sets, A and B, where A = {2, 3, 4, 5} and B = {5, 4, 3, 2}, they are said to be equal.

When two sets contain the same number of elements even when the elements are different, they are considered equivalent. For example, A = {1,2,3,4} and B = {9, a, 3, w} are equivalent.

Two sets are considered disjoint if they do not contain a common element. For example, sets A and B are disjoint if A = {1, 2, 3, 4} and B = {7, 8, 9, 10}.

Set A is considered a subset of B if all elements of A are present in set B. It is expressed mathematically by the notation A ⊆ B. By this definition, sets are considered subsets of themselves. For example, if B = {4, 6, 8,} and A = {6, 8}, A ⊆ B. When a set (A) is not a subset of another (B), it is denoted by A ⊈ B .

Empty sets are also regarded as subsets to every set. And empty sets have one subset, itself, whilst non-empty sets have at least 2 subsets, 0 and itself.

If A ⊆ B, yet A ≠ B, then A is considered a proper subset of B. This can be denoted by A ⊂ B. For example, if A = {9, 12} and B = {3, 6, 9, 12}, then A ⊂ B.

Set A is considered a superset of B if all elements of B are present in set A. it is denoted by the symbol ⊇. For example, if A = {1,2,3,4} and B = {1,2,3}, then A ⊇ B.

This is a set which contains elements of all related sets without repetition. It is denoted by the symbol U. For example, if A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then the universal set here is U = {1, 2, 3 , 4, 6, 8}.

Under certain conditions, operations of sets can be carried out in set theory. Some basic operations are:

Union of sets

Intersection of sets

Complement of a set

Cartesian product of sets

Set difference

A union of sets contains all elements of the related sets. So if we have set A and B, a union will be all elements of A and B. It is denoted by the symbol U. Mathematically, a union of A and B will look like AUB.

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, AUB = {1, 2, 3, 4, 5, 6}. And this can be represented on a Venn diagram below.

An intersection set is one that contains common elements of related sets. An intersection of sets A and B will be elements that appear in both A and B. It is denoted by the symbol ∩. This means an intersection of sets A and B will mathematically be written as A ∩ B.

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, A ∩ B = {3, 4}. This can also be represented on a Venn diagram below.

Complement sets contain all elements in the universal set that are not in the given set. Assuming A is a subset of a much larger set called a universal set, the complement of A is all elements present in the universal set that aren't present in A. The complement will be denoted by A '.

If we have U = {2, 4, 6, 8, 10} and A, the subset of U is = {4, 6, 8}. Then A '= {2, 10}.

The Cartesian Product of sets is defined as the set of all ordered pairs (x, y) from two sets, A and B such that x belongs to A and y belongs to B.

If A = {1, 2} and B = {3, 4, 5}, then the Cartesian Product of A and B is {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}, and this is noted by .

Set difference lists the elements in set A that are not present in set B. It is denoted by A - B. For example, if A = {1, 2, 3, 4} and B = {1, 3, 5, 7 }, then A - B = {2, 4}.

Sets, just like numbers, also have properties associated with them. The set formula is given in general as n (A∪B) = n (A) + n (B) - n (A⋂B), where A and B are two sets and n (A∪B) shows the number of elements present in either A or B and n (A⋂B) shows the number of elements present in both A and B. We will discuss six important properties given three sets A, B and C in this section.

**Commutative property:**

AUB = BUA

A ∩ B = B ∩ A

**Associative property:**

(A ∩ B) ∩ C = A ∩ (B ∩ C)

(AUB) UC = AU (BUC)

**Distributive property:**

AU (B ∩ C) = (AUB) ∩ (AUC)

A ∩ (BUC) = (A ∩ B) U (A ∩ C)

**Identity property:**

AU ∅ = A

A ∩ U = A

**Complement property:**

AUA '= U

**Idempotent property:**

A ∩ A = A

AUA = A

Here are a few worked examples on sets.

Define the following sets in the Venn Diagram

A ∩ B

B '

A ⋃ B

Answer:

A ∩ B means elements that are present in both A and B. That is where they both intersect.

A ∩ B = {5, 4}.

B 'means all elements that are not present in B.

B '= {1, 2, 7, 8}.

A ⋃ B means all elements that appear in either A or B.

A ⋃ B = {2, 4, 5, 6, 8, 10, 14}

Let A = {12, 13, 15, 17, 18, 19}, and B = {13, 14, 16, 18, 19, 21, 25}

Find

A ∩ B

A ⋃ B

The cardinality of B

Answer:

A ∩ B = {13, 18, 19}

A ⋃ B = {12, 13, 14, 15, 16, 17, 18, 19, 21, 25}

n (B) = 7

- Sets in mathematics are an organised collection of objects called elements.
- Sets can be represented in semantic form, roster form, and set builder form.
- The roster form representation of sets is denoted by the curly braces and separated by commas.
- Every example of a set has an empty set as a subset, and empty sets are denoted by either {} or ∅.
- A set A is considered a subset of B if all elements of A are present in set B.
- An intersection set is one that contains common elements of related sets.
- Complement sets are sets that contain all elements in the universal set that are not in the given set.
- A union of sets contains all elements of the related sets.
- Cardinality denotes the total number of elements in a set.

They are a collection of numbers related to a topic.

These are the set of all variables that makes an equation true.

More about Sets Math

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