# Sets

Last Updated: April 29, 2024

## 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

1. 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.
2. 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.
3. 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.
4. Singleton Set: A set that contains only one element. For example, the set containing only the number 7, {7}, is a singleton set.
5. 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}.
6. Power Set: The set of all subsets of a given set. For example, the power set of {1,2} is {∅,{1},{2},{1,2}}
7. Disjoint Sets: Two sets that have no elements in common. For instance, {1,2,3} and {4,5,6} are disjoint sets.
8. 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.
9. 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}.
10. 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.
11. 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

1. Natural Numbers (N): The set of positive integers starting from 1. It’s represented as{1,2,3,…}.
2. Whole Numbers (W): The set of non-negative integers, including 0. It’s represented as {0,1,2,3,…}.
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,…}.
4. Rational Numbers (Q): The set of numbers that can be expressed as a fraction p/q, where p and q are integers and q0.
5. Irrational Numbers (′Q′): The set of numbers that cannot be expressed as a rational number, such as 2​ or π.
6. 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:

1. 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}.
2. 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 AB={3}.
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}.
4. 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}.
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}.
6. 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

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

## Representation of Sets in Set Theory

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

1. 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}.
2. 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}.
3. 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}.

## 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

1. 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.
2. Question: What is the difference between a union and an intersection of two sets?
Answer: The union of two sets A and B (denoted AB) consists of all elements that are in either set or both. The intersection of two sets A and B (denoted AB) includes only the elements that are in both sets.
3. 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}.
4. 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}.
5. 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)}.
6. Question: What is the symmetric difference between two sets 𝐴 and 𝐵, and how is it represented?
7. 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:

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

1. Intersection 𝐴∩𝐵∩𝐶: The intersection includes elements that are common to all three sets:𝐴∩𝐵∩𝐶={5}
2. 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}
3. 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}

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

1. Elements of A and B: 𝐴={1,2,3,4,5,6,7,8,9,10}, 𝐵={5,6,7,8,9,10,11,12,13,14,15}
2. 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
3. Cardinality of the Union 𝑛(𝐴∪𝐵)n(AB): 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

## 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.

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