## Sets

## What is Set?

**A set is a collection of well-defined and distinct objects or elements.** For instance, the collection of vowels forms a set, as they are clearly defined as “a, e, i, o, u.” Similarly, whole numbers and prime numbers are examples of sets, as their members are uniquely identifiable and consistent.

## Types of Sets

**Finite Set:**A set with a specific number of elements. For example, the set of vowels in the English alphabet, {𝑎,𝑒,𝑖,𝑜,𝑢}, is a finite set with five elements.**Infinite Set:**A set that has an unlimited number of elements. An example is the set of natural numbers, {1,2,3,4,…}, which goes on infinitely.**Empty Set (Null Set):**A set that has no elements. It’s denoted by ∅ or {}. For instance, the set of even numbers that are also odd, which is ∅ because no number can be both.**Singleton Set:**A set that contains only one element. For example, the set containing only the number 7, {7}, is a singleton set.**Subset:**A set that consists of elements all drawn from another set. For example,{1,2} is a subset of the set {1,2,3,4}.**Power Set:**The set of all subsets of a given set. For example, the power set of {1,2} is {∅,{1},{2},{1,2}}**Disjoint Sets:**Two sets that have no elements in common. For instance, {1,2,3} and {4,5,6} are disjoint sets.**Equal Sets:**Two sets that have the same elements, regardless of order or repetition. For example,{a,b,c} and {c,b,a} are equal sets.**Complementary Sets:**For a given universal set U and a subset A of U, the complement of A (denoted as Ac) consists of all elements in U that are not in A. For instance, if U={1,2,3,4,5} and A={1,3,5}, then the complement of A is Aᶜ={2,4}.**Ordered Sets:**Sets where the order of elements matters. For example, the set {1,2,3} is different from the set {3,2,1} if they’re considered as ordered sets.**Multiset:**A set where elements can repeat. For instance, a multiset could be represented as {a,a,b,c}, indicating two instances of the letter ‘a’.

## Sets in Maths Examples

**Natural Numbers (N):**The set of positive integers starting from 1. It’s represented as{1,2,3,…}.**Whole Numbers (W):**The set of non-negative integers, including 0. It’s represented as {0,1,2,3,…}.**Integers (Z):**The set of all positive and negative whole numbers, including 0. It’s represented as{…,−3,−2,−1,0,1,2,3,…}.**Rational Numbers (Q):**The set of numbers that can be expressed as a fraction*p*/*q*, where*p*and*q*are integers and*q***≠**0.**Irrational Numbers (′Q′):**The set of numbers that cannot be expressed as a rational number, such as**√**2 or*π*.**Real Numbers (R):**The set that encompasses both rational and irrational numbers, represented as ′Q∪Q′. It covers all points on the number line.

## Operations on Sets

Operations on sets are fundamental ways to manipulate and combine sets, providing powerful tools in mathematics and computer science. Here are key set operations:

**Union:**The union of two sets A and B (denoted A∪B) includes all elements that are in A, B, or both. For example, if A={1,2,3} and B={3,4,5}, then A∪B={1,2,3,4,5}.**Intersection:**The intersection of two sets A and B (denoted A∩B) consists of all elements that are in both A and B. For example, if A={1,2,3} and B={3,4,5}, then*A*∩*B*={3}.**Difference:**The difference of two sets A and B (denoted A−B) includes elements that are in A but not in B. For example, if A={1,2,3} and B={3,4,5}, then A−B={1,2} And B−A={4,5}.**Symmetric Difference:**The symmetric difference of two sets A and B (denoted A△B) consists of elements that are in A or B, but not both. For example, if A={1,2,3} and B={3,4,5}, then A△B={1,2,4,5}.**Complement:**The complement of a set A (denoted Aᶜ) consists of all elements that are not in A but are in the universal set U. For example, if U={1,2,3,4,5} and A={1,2,3}, then Ac={4,5}.**Cartesian Product:**The Cartesian product of two sets A and B (denoted A×B) consists of all ordered pairs (a,b) where a∈A and b∈B. For example,A={1,2} and 𝐵={3,4}, then 𝐴×𝐵={(1,3),(1,4),(2,3),(2,4)}.

## Sets Formulas

Formula | Description |
---|---|

(A∪B)∪C=A∪(B∪C) | Associative Law (Union): Union operation is associative, meaning grouping order doesn’t matter. |

(A∩B)∩C=A∩(B∩C) | Associative Law (Intersection): Intersection operation is associative. |

A∪(B∩C)=(A∪B)∩(A∪C) | Distributive Law (Union over Intersection): Union distributes over intersection. |

A∩(B∪C)=(A∩B)∩(A∩C) | Distributive Law (Intersection over Union): Intersection distributes over union. |

(A∪B∪C)ᶜ=Aᶜ∩Bᶜ∩Cᶜ | De Morgan’s Law (Union): The complement of a union is the intersection of the complements. |

(A∩B∩C)ᶜ=Aᶜ∪Bᶜ∪Cᶜ | De Morgan’s Law (Intersection): The complement of an intersection is the union of the complements. |

A∪B=B∪A | Commutative Law (Union): Union operation is commutative. |

A∩B=B∩A | Commutative Law (Intersection): Intersection operation is commutative. |

Here are some key formulas and properties involving three sets *A*, *B*, and *C*:

1.**Associative Law (Union):** It indicates that the union operation is associative, meaning the grouping order of the sets does not matter.

(A∪B)∪C=A∪(B∪C)

2. **Associative Law (Intersection):** It indicates that the intersection operation is associative.

(A∩B)∩C=A∩(B∩C)

3. **Distributive Law (Union over Intersection):** It shows how union distributes over intersection.

A∪(B∩C)=(A∪B)∩(A∪C)

4. **Distributive Law (Intersection over Union):** This shows how intersection distributes over union.

A∩(B∪C)=(A∩B)∩(A∩C)

5. **De Morgan’s Law (Union):** This shows the complement of a union is the intersection of the complements.

(A∪B∪C)ᶜ=Aᶜ∩Bᶜ∩Cᶜ

6. **De Morgan’s Law (Intersection):** This shows the complement of an intersection is the union of the complements.

(A∩B∩C)ᶜ=Aᶜ∪Bᶜ∪Cᶜ

7. **Commutative Law (Union):** This indicates that the union operation is commutative.

A∪B =B∪A

8. **Commutative Law (Intersection):** This indicates that the intersection operation is commutative.

A∩B=B∩A

Formula Expression | Description |
---|---|

n(A∪B)=n(A)+n(B)−n(A∩B) | Calculates the number of elements in the union of two sets A and B by adding their individual sizes and subtracting the size of their intersection. |

n(A∪B)=n(A)+n(B) <br> If 𝐴∩𝐵=∅ | If sets A and B are disjoint (no common elements), the size of their union is simply the sum of their sizes. |

n(A−B)+n(A∩B)=n(A) | The size of set A can be determined by adding the size of the difference of A and B to the size of their intersection. |

n(B−A)+n(A∩B)=n(B) | Similarly, the size of set B is the sum of the size of the difference of B and A and the size of their intersection. |

n(A−B)+n(A∩B)+n(B−A)=n(A∪B) | The total number of elements in the union of A and B can be found by adding the sizes of the differences and their intersection. |

n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C) | This formula calculates the number of elements in the union of three sets A, B, and C by considering all possible intersections to avoid over-counting elements present in multiple sets. |

## Properties of Sets

Commutative Property :A∪B = B∪A A∩B = B∩A |

Associative Property : A ∪ ( B ∪ C) = ( A ∪ B) ∪ C A ∩ ( B ∩ C) = ( A ∩ B) ∩ C |

Distributive Property :A ∪ ( B ∩ C) = ( A ∪ B) ∩ (A ∪ C) A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C) |

De morgan’s Law :Law of union : ( A ∪ B )’ = A’ ∩ B’ Law of intersection : ( A ∩ B )’ = A’ ∪ B’ |

Complement Law :A ∪ A’ = A’ ∪ A =UA ∩ A’ = ∅ |

Idempotent Law And Law of a null and universal set :For any finite set A A ∪ A = A A ∩ A = A ∅’ = U ∅ = U’ |

## Representation of Sets in Set Theory

In set theory, various notations are used to represent sets, each offering a unique way to list elements:

**Semantic Form:**This notation describes sets by their defining properties. For example, the set of natural numbers greater than 3 can be written as {𝑥∣𝑥∈𝑁 and 𝑥>3}.**Roster Form:**This notation lists all elements of a set explicitly, separated by commas and enclosed in curly brackets. For instance, the set of vowels can be represented as {𝑎,𝑒,𝑖,𝑜,𝑢}.**Set Builder Form:**This notation defines a set by a rule or condition that its elements satisfy. For example, the set of even integers can be written as {𝑥∣𝑥 is an integer and 𝑥%2=0}.

Set of first five even natural numbers | ||
---|---|---|

Semantic Form | Roster Form | Set Builder Form |

A set of first five even natural numbers | {2, 4, 6, 8, 10} | {x ∈ ℕ | x ≤ 10 and x is even} |

## Visual Representation of Sets Using Venn Diagram

A **Venn Diagram** is a visual representation of sets, with each set shown as a circle. The elements of a set are placed inside its respective circle. Often, the circles are enclosed by a rectangle, which represents the universal set. Venn diagrams illustrate how sets relate to one another by showing intersections, unions, and differences, making them a powerful tool for understanding set operations and relationships.

## Sets Symbols

Symbol | Meaning |
---|---|

∈ | Indicates membership of an element in a set. For example, 𝑎∈𝐴 means 𝑎a is an element of set A. |

∉ | Indicates that an element is not a member of a set. For example, 𝑏∉𝐵 means b is not in set B. |

⊆ | Indicates that set A is a subset of set B (i.e., all elements of A are in B). |

⊂ | Indicates that set A is a proper subset of set B (i.e., all elements of A are in B and B has other elements not in A). |

∩ | Represents the intersection of two sets, which includes elements common to both sets. |

∪ | Represents the union of two sets, which includes all elements that are in either set. |

− | Represents the difference between two sets A and B, including elements that are in A but not in B. |

△ | Represents the symmetric difference between two sets, including elements that are in one set or the other, but not both. |

∅ | Represents the empty set, which contains no elements. |

× | Represents the Cartesian product of two sets, which consists of all ordered pairs formed by elements from both sets. |

𝐴𝑐 | Represents the complement of set A, which includes all elements not in A but in the universal set U. |

**Question:**Define a set and give an example of a finite set.**Answer:**A set is a collection of distinct and well-defined objects. An example of a finite set is the set of vowels in the English alphabet: {*a*,*e*,*i*,*o*,*u*}, which contains five elements.**Question:**What is the difference between a union and an intersection of two sets?**Answer:**The union of two sets*A*and*B*(denoted*A*∪*B*) consists of all elements that are in either set or both. The intersection of two sets*A*and*B*(denoted*A*∩*B*) includes only the elements that are in both sets.**Question:**How do you represent a set using set builder notation, and give an example?**Answer:**Set builder notation defines a set by specifying a condition or property that its elements must satisfy. An example is the set of even integers, which can be represented as {𝑥∣𝑥 is an integer and 𝑥%2=0}.**Question:**Explain the complement of a set and provide an example.**Answer:**The complement of a set*A*(denoted 𝐴ᶜ) consists of all elements not in 𝐴 but present in the universal set*U*. For instance, if*U*={1,2,3,4,5} and 𝐴={1,3,5}, then 𝐴ᶜ={2,4}.**Question:**What is the Cartesian product of two sets*A*and*B*and how is it represented?**Answer:**The Cartesian product of two sets*A*and*B*(denoted 𝐴×𝐵) consists of all ordered pairs (*a*,*b*) where 𝑎∈𝐴 and 𝑏∈𝐵. For example, if 𝐴={1,2} and 𝐵={3,4}, then 𝐴×𝐵={(1,3),(1,4),(2,3),(2,4)}.**Question:**What is the symmetric difference between two sets 𝐴 and 𝐵, and how is it represented?**Answer:**The symmetric difference between two sets 𝐴 and 𝐵(denoted 𝐴△𝐵) includes elements that are in one set or the other, but not both. For example, if 𝐴={1,2,3}= and 𝐵={3,4,5}, then 𝐴△𝐵={1,2,4,5}.

### Question 1:

Given three sets *A*,*B*, and *C* such that 𝐴={1,2,3,4,5}, 𝐵={3,4,5,6,7}, and 𝐶={5,6,7,8,9}, answer the following:

**Find 𝐴∩𝐵∩𝐶**(the intersection of the three sets).**Calculate 𝐴∪𝐵∪𝐶**(the union of the three sets).**Determine (𝐴−𝐵)∪(𝐵−𝐴)**(the symmetric difference between 𝐴 and 𝐵).

#### Answer:

**Intersection 𝐴∩𝐵∩𝐶:**The intersection includes elements that are common to all three sets:𝐴∩𝐵∩𝐶={5}**Union 𝐴∪𝐵∪𝐶:**The union includes all elements that are in any of the three sets:𝐴∪𝐵∪𝐶={1,2,3,4,5,6,7,8,9}**Symmetric Difference (𝐴−𝐵)∪(𝐵−𝐴):**This includes elements that are in 𝐴 but not in 𝐵 and vice versa:(𝐴−𝐵)∪(𝐵−𝐴)={1,2,6,7}

### Question 2:

Let sets 𝐴 and 𝐵 be defined as follows:

𝐴={𝑥∣𝑥 is an integer, and 1≤𝑥≤10}

𝐵={𝑥∣𝑥 is an integer, and 5≤𝑥≤15}

**Find the elements of sets 𝐴 and 𝐵**.**Determine 𝑛(𝐴∩𝐵)**(the cardinality of the intersection of 𝐴 and 𝐵).**Find 𝑛(𝐴∪𝐵)**(the cardinality of the union of 𝐴 and 𝐵).

#### Answer:

**Elements of**𝐴={1,2,3,4,5,6,7,8,9,10}, 𝐵={5,6,7,8,9,10,11,12,13,14,15}*A*and*B*:**Cardinality of the Intersection 𝑛(𝐴∩𝐵):**The intersection consists of elements common to both sets:𝐴∩𝐵={5,6,7,8,9,10}Therefore, the cardinality of this set is:𝑛(𝐴∩𝐵)=6**Cardinality of the Union 𝑛(𝐴∪𝐵)**The union includes all unique elements in both sets:𝐴∪𝐵={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}Therefore, the cardinality of this set is: 𝑛(𝐴∪𝐵)=15*n*(*A*∪*B*):

## FAQ’s

### What is the formula for sets?

The formula 𝑛(𝐴∪𝐵)=𝑛(𝐴)+𝑛(𝐵)−𝑛(𝐴∩𝐵) calculates the number of elements in the union of two sets by adding their individual sizes and subtracting the size of their intersection, avoiding double counting.

### What is a set in maths?

In mathematics, a set is a collection of distinct and well-defined elements, such as numbers, letters, or symbols. Sets are fundamental to various mathematical concepts and operations, providing a structure for organizing and manipulating elements in disciplines like algebra and logic.

### Why is the null set called a set?

A null set, or empty set, is called a set because it adheres to the definition of a set—a collection of distinct, well-defined elements. In this case, it is a set with no elements, but still conforms to set properties.

### What is *z* in sets?

In set notation, Z denotes the set of all integers, encompassing positive, negative, and zero integers. This notation is derived from the German word “Zahlen,” meaning numbers, and is widely used in mathematical contexts.

### What is a unit set?

A unit set, also known as a singleton set, contains only one element. For example, the set *A*={7} is a unit set because it has a single, distinct element, which makes it unique compared to sets with multiple elements.