Types of Sets

Introduction

Sets are essentially a collection of different items that constitute a group in mathematics. A set can contain any number of elements, such as numbers, days of the week, car types, and so on. An element of the set refers to each object in the set. When writing a set, curly brackets are utilized. This is an example of a set in its most basic form. Set A consists of the numbers 1,2,3,4,5. A set of items can be represented using a variety of notations. A roster form or a set builder form are commonly used to represent sets. 

Let’s take a closer look at each of these terms.

Sets Definition

A set is a systematic collection of objects in mathematics. A capital letter is used to name and represent sets. The elements that make up a set in set theory can be anything: humans, shapes, letters of the alphabet, numbers, variables, and so on.

Sets in Math 

Examples

A collection of even natural numbers smaller than ten is defined, but a collection of bright pupils in a class is not. As a result, a set A = 2, 4, 6, 8 can be used to represent a collection of even natural numbers less than 10.

In mathematics, various sorts of relations define the link between sets. There are eight different types of relations in Math. 

• Empty Relation

If no element of set A is related or mapped to any element of A, then the relation R in A is an empty relation, i.e, R = Φ. 

For example, set A consists of only 50 apples in a box. 

Is there any possibility of finding a relation R of getting any stone in the box? 

No! R is a void or empty relation since there are only 50 apples and no stones.

• Universal Relation

A relation R in a set, say X is a universal relation if each element of X is related to every element of X, i.e., R = X × X. Also called Full relation. 

For example, X={1,3,5,7,…} i.e set of all odd numbers and Y is a set of all natural numbers. The relation between X and Y is universal as every element of set A is in set B.

• Identity Relation

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}. 

For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). 

If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

• Inverse Relation

Inverse relation is seen when a set has elements which are inverse pairs of another set. 

For example if set A = {(p, q), (r, s)}, then inverse relation will be R-1= {(q, p), (s, r)}.

• Reflexive Relation

In a reflexive relation, every element maps to itself. 

For example, consider a set X = {2, 4}. Now an example of reflexive relation will be R = {(2, 2), (4, 4), (2, 4), (4, 2)}.

• Symmetric Relation

In a symmetric relation, if a=b is true then b=a is also true. In other words, a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. 

For example, the symmetric relation will be R = {(2, 4), (4, 2)} for a set A = {1, 2}. 

• Transitive Relation

A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

• Equivalence Relation

When a relation is reflexive, symmetric and transitive simultaneously, it is known as an equivalence relation.

Elements of a Set

The components that make up a set are referred to as elements or members of the set. Curly brackets contain the elements of a set, which are separated by commas. The ∈ sign is used to indicate that an element is part of a set. In the preceding example, 2 ∈ A. The symbol is used to represent an element that is not a member of a set. Here, 3 ∉ A.

Types of Sets

There are several different types of sets in mathematics. Singleton, finite, infinite, empty, and other terms are among them.

• Singleton Sets

A singleton set, also known as a unit set, is a set with only one element. Set A = {k | k is an integer between 3 and 5}, resulting in set A = {4}.

• Finite Sets

A finite set is a set with a finite or countable number of items, as the name implies. Set B ={ k | k is a prime number smaller than 20}, for example, is B = 2,3,5,7,11,13,17,19.

• Infinite Sets

An infinite set refers to the set that includes an unlimited number of items. Set C ={k|k is a multiple of 3} as an example.

• Empty or Null Sets

An empty set, also known as a null set, is a set that has no elements. The symbol ∅ is used to represent an empty set. It’s pronounced ‘phi.’ Set X = {} or ∅ is an example.

• Equal Sets

Equal sets are made up of two sets that have the same components in them. If  A = {1,2,3} and B = {1,2,3} are two sets, then the sets A and B are the same in this case. A = B is one way to express this.

• Unequal Sets

Unequal sets are one of the popular types of sets in mathematics that have at least one element that is different.If A = {1,2,3} and B = {2,3,4} are two sets, then the sets A and B aren’t equal in this situation. A ≠ B is a way to express this.

• Equivalent Sets

When two sets contain the same number of elements, but different elements, then they are said to be equivalent sets. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

• Overlapping Sets

If at least one element from set A appears in set B, the two sets are said to overlap. Example: A = {2,4,6} B = {4,8,10}. Element 4 appears in both sets A and B in this case. As a result, A and B are two sets that overlap.

• Disjoint Sets

If there are no shared elements in both sets, then they are disjoint sets. For instance, A = {1,2,3,4} and B = {5,6,7,8}. Sets A and B are disjoint in this case.

Subset and Superset

If every member in set A is also present in set B, set A is a subset of set B (A ⊂ B) and set B is the superset of set A (B ⊃ A).

Example: A = {1,2,3} B = {1,2,3,4,5,6}

Set B is the superset of set A, as denoted by B ⊃ A.

Universal Set

A universal set is a collection of all items related to a specific topic. The letter ‘U’ stands for the universal set. Let U stand for “the list of all road transport vehicles.” This universal set includes a set of automobiles, a set of cycles, and a set of trains, all of which are subsets of this universal set.

Power Sets

The collection of all subsets that a set might contain is called a power set. Set A = {1,2,3} as an example. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Conclusion 

We’ve covered set definitions, types of sets, and types of relation in sets, as well as examples and more.