Elements in Sets

A group of objects is referred to as a set. The elements or members of a set are the objects that make up the set. A set’s elements can be any kind of object, including sets! A set’s members don’t even have to be the same kind. A set, for example, can be made up of numbers and names, even if they have no practical use.

Element

The objects that make up a set are known as its element or members.

The components of a set are typically written inside a pair of curly braces and denoted by commas. We normally use capital letters to symbolise sets, such as A, B, C, S, and T, and lowercase letters to signify their generic elements, such as a, b, c, s, and t, respectively. We use the notation b,B to signify that b  B, which implies ” b belongs to B ” or ” b is a member of B “.

For example: A={a,b,c,s,t}

The set whose elements (members) are a, b, c, s, and t is referred to as ‘A’.

Element in set

Elements in a set can be represented by many types and methods. Some are pre-defined sets which already have a notation. Some examples are as follows-

. N:set of natural numbers {1,2,3,…}

. Z:set of integers {…,-2,-1,0,1,2,…}

. R:set of real numbers: The real numbers are the union or combination of rational and irrational numbers. Positive real numbers represent positions to the right of the origin, while negative real numbers represent locations to the left.

. Q:set of rational numbers: The rational numbers are those that have the form ab, where a and b are integers and b≠0. Every integer is a rational number since b can equal 1.

Because they all have an infinite number of items, they are all infinite sets. Finite sets, on the other hand, have a finite number of items.

Set order

The number of elements in a set is determined by its order. It is a term that describes the size of a set. The cardinality of a set is another name for its order.

The size of a set, whether finite or infinite, is referred to as set of finite order or set of infinite order, respectively.

Objects in set

Elements are everything for a set. It determines whether the type, name and function of a set will be.

Null set

An empty set, also known as a void set or null set, is a set that has no elements. It’s indicated by the letters { } or Ø.

Finite set

A finite set is a collection of elements with a fixed number of elements. Ex- X = {1,2,3,4,5,}

Singleton Set

A singleton set is a set that only contains one element. Ex- X = {1}

Infinite set

The term “infinite set” refers to a set that is not finite. Ex- X = {1,2,3,4,5,6,7,8,9……}

Equal sets

The two sets X and Y are said to be equal if they contain the exact identical elements, regardless of their order. Ex- X = {1,2,3,4} and Y = {4,3,2,1}, then X = Y.

Equivalent/ analogous set

When the number of items in two different sets is the same, they are referred to as analogous sets. The order of the sets is irrelevant in this case. Ex- If X = {1,2,3,4} and Y = {Red, Blue, Green, Black}

There are four objects in set X, and there are four objects in set Y as well. As a result, sets X and Y are Equivalent/ analogous.

Disjoint Sets

If the two sets X and Y do not share any elements, they are said to be disjoint. Ex- X = {1,2,3,4} and set Y = {5,6,7,8} are disjoint sets.

Subset

If every element of X is also an element of Y, the set ‘X’ is said to be a subset of Y, denoted as X  Y. Ex- X = {1,2,3} and Y={1,2}, Then {1,2} X.

Superset

If all of the components of set Y are also elements of set X, then set X is said to be the superset of Y. X Y is the symbol for it. Ex- set X = {1, 2, 3, 4} and set Y = {1, 3, 4}.

Proper Subset

If X is the proper subset of Y and X is not equal to Y, then X is written as XY. Ex- X = {2,5} is a subset of Y = {2,5,7} and is a proper subset also.

Universal Set

The universal set is a set that contains all the sets applicable to a given condition.

Above written every type of set is the result of how objects in a set are behaving. Alone on the number of elements can differ the type of a set. Further, similarity and equality of number of elements changed the type of the set

Conclusion

In this article we learned that Sets are used to hold a group of objects that are connected. They are significant in every branch of mathematics because sets are used or referred to in some way in every field of mathematics. They are necessary for the construction of more complicated mathematical structures. It is vital in our daily lives since it is used to group things together and count whether or not an object is part of a set. When buying groceries, for example, you can observe how the things are organised, such as in the first case, soap, the second case, foods, and so on.