The set concept in mathematics is about representing bulk data in groups based on certain characteristics. This blog post will help students learn about types of sets and their operations through definitions, examples, and diagrams
Sets: The concept of sets in mathematics is an interesting way of grouping elements using analytical and logical thinking. Learning the concept of sets will help your child solve math problems, analyze data, or even categorize items in everyday life.
Here, we are going to discuss in detail the definition of sets, types, and operation of sets with visual representations that clarify how collections of objects or numbers relate to each other.
Definition of Sets
A set is a collection of items in a well-defined way, where each item shares some common properties and appears only once in a set. These objects, included in a set, are called elements. A set is usually denoted by a capital letter (A, B, S, T, etc.), and the elements are enclosed within curly brackets ({ }).
For example,
A = {2, 3, 5, 7} is a set of all prime numbers between 2 and 10.
B = {a, e, i, o, u} is a set of all vowels.
This form of representing a set is also called the Roaster form.
Also read: Volume - Formula, Definition, Calculate, Examples
Elements of a Set
The elements in a set are written within the curly brackets, separated by commas. The elements have some important criteria as mentioned below:
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Well-defined: The elements or members of a set are well-defined, which means they have some specific characteristics, not anything vague or subjective.
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Common properties: All the elements of a set must share a common property that defines the set’s characteristics.
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Single entity: Each element in a set occurs only once, and there is no repetition of any particular item.
For example, a set of “all students who scored above 80% in the final exam of class 5” is a well-defined set with a common characteristic and contains students’ names without any repetition.
Basic Sets Examples
In mathematics, your child may encounter some common or basic sets of elements as listed below:
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Set of natural numbers: N = {1, 2, 3, ...}
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Set of whole numbers: W = {0, 1, 2, 3, ...}
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Set of integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
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Set of prime numbers: P = {2, 3, 5, 7, 11, …}
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Set of even numbers: E = {2, 4, 6, 8, …}
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Set of odd numbers: O = {1, 3, 5, 7, …}
Also read: Sequences and Series - Definitions, Types, Concepts
Types of Sets
Sets can be categorized into different types based on their characteristics. Let’s find out what the different types of sets are:
Singleton Set
It is a type of set with only one element contained in it. It is also called a unit set.
For example,
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A set of even prime numbers is A = {2}. It contains only one element, 2.
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A set of integers between -1 and 1 is B = {0}. It contains only one element, 0.
Empty Set
It is a set that doesn’t contain any items. It is also called a null set. An empty set is denoted using the symbol '∅' (phi).
For example,
Finite Set
This set has a finite and countable number of elements.
For example,
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A set of odd numbers between 1 and 10 is B = {1, 3, 5, 7, 9}. It contains a finite number of elements.
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The set of different colors of a rainbow is also a finite set containing 7 elements.
Infinite Set
A set that contains an infinite number of elements is called an infinite set.
For example,
Equal Set
Equal sets are those sets which contain the same elements and same number of elements.
For example,
A = {a, b, c, d} and B = {c, d, a, b} are equal sets.
Equivalent Set
Two sets having an equal number of elements are called equivalent sets, even if the elements are different.
For example,
The set of vowels V = {a, e, i, o, u} and the set of odd numbers between 1 and 10, P = {1, 3, 5, 7, 9}, are equivalent sets because each of them contains 5 elements.
Overlapping Sets
Two sets are called overlapping when at least one element is common between them.
For example,
A set of factors of 12 is A = {1, 2, 3, 4, 6, 12}, and a set of 15 is B = {1, 3, 5, 15}. Here, A and B are overlapping sets because the elements 1 and 3 are common between the two sets.
Disjoint Sets
These are the sets that have no common elements between them.
For example,
A set of even numbers and a set of odd numbers have no common element between them. So, they are disjoint sets
Subset and Superset
A set is called a subset of another set if all the elements of the first set are also present in the second set.
For example,
If M = {10, 20, 30} and N = {5, 10, 15, 20, 25, 30}, then M is called a subset of N.
It can be denoted as M ⊆ N.
In this case, N is called the superset of M. It can be denoted as N ⊇ M.
Universal Set
A universal set is the set that includes all elements of the rest of the sets. In other words, all sets are subsets of one universal set.
For example,
If there are three sets as A = {1, 3, 5}, B = {2, 4, 6}, and C = {1, 2, 3, 6}, then the universal set U = {1, 2, 3, 4, 5, 6}.
Power Set
A set of all subsets is called the power set. The power set of any set contains the following sets as its elements:
For example,
If A = {x, y, z} is a set, then the power set of A contains the elements ∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, and {x, y, z}.
Therefore, P = {∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.
Also read: Simple Equations: Definition, Transposition and Uses
Operation of Sets
A number of operations can be conducted involving the sets. A brief explanation of each of these sets operations is given below
Union of Sets
A union of sets refers to the generation of a set that contains all the elements of the individual sets.
If A and B are two sets, then the union of A and B is denoted by A U B.
For example, if A = {1, 2, 3, 4} and B = {2, 4, 6}, then A U B = {1, 2, 3, 4, 6}.
Intersection of Sets
The intersection of sets includes all elements that are common to both sets.
If A and B are two sets, then the intersection of A and B is denoted by A ∩ B.
For example, if A = {1, 2, 3, 4} and B = {2, 4, 6}, then A ∩ B = {2, 4}.
Difference of Sets
The difference of two sets means the set of elements that belongs to the one set but not to the other set.
If A and B are two sets, then the difference of A and B is denoted by A - B.
For example, if A = {1, 2, 3, 4} and B = {2, 4, 6}, then A - B = {1, 3}.
Complement Sets
The set complement for a set A is denoted by A’, which includes all elements of the universal set (U), except the elements contained in A. In other words, A’ can be denoted by U - A.Also Read: Skip Counting
Also Read: Skip Counting
Sets Venn Diagram
A Venn diagram is the diagrammatic representation of sets, their relationships, and operations performed with them. It typically uses circles to represent the sets. The circles are drawn within a large rectangle, which denotes the universal set.
If A and B are two different sets, we can represent their relationships and operations through the Venn diagram.
Sets Solved Examples
1. Explain the relationships between the following sets.
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M = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
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A = {0, 1, 2, 3}
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C = {1, 3, 5, 7, 9, 11, 13}
Solutions:
2. If A = {11, 13, 17}, B = {10, 11, 12, 13, 14, 15}, and C = {5, 10, 15, 20}
Find A ∩ B and B ∩ C. What is the relationship between A and C?
Solution:
A ∩ B is the intersection of A and B. So, A ∩ B = {11, 13}.
B ∩ C is the intersection of B and C. So, B ∩ C = {10, 15}.
A and C are disjoint sets because they have no elements in common.
Sets are an important concept in mathematics and also many other scientific applications. Understanding the types of sets and operations performed on them will help your child to analyze and solve problems involving sets of numbers and objects easily and efficiently.
Also Read: Types of Line in Math
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