What are sets and why are they used in mathematics?

The idea of sets is quite important in the field of mathematics. The concept of sets is used in major branches such as Physics, Chemistry, Engineering and has been recently used in medical departments too. The concept of the set has been introduced by the famous mathematician, Georg Cantor who majorly stated that a set represents many values but is taken as one. 

So a set can be defined as a collection of well-defined several objects taken as one. For example, a collection of odd numbers can be called a set. Here, 1 is taken as a value but does not include 2 in the set. Similarly, a collection of kitchen utensils is not a set since it has no defined criteria.

Real-life examples of a set

  • A collection of carrom men in the game of carrom.
  • Names of all the available seasons in a year.
  • A collection of things is present in a wallet. e.g coins, notes, etc.

So, the objects present within a set are termed elements of the set or members.

Let us assume that N represents the set of odd numbers 1, 3, 5, 7, 9…. Hence we can say that 1, 3, 5 are the elements of set N. Here 5 is a member of the set N, so we can write 5 ∈ N which means 5 belongs to the set N or is a member of it. Similarly, we can say 2 is not a member of the set N. Thus, we write it as 2 ∉ N which means 2 does not belong to the set N or is not a member of the set N.

Various important sets in mathematics 

Sets can be divided into major groups which are as follows:

  • Set consisting of Natural Numbers: N = { 1, 2, 3, 4, 5, 6, 7….}
  • Set consisting of Whole Numbers: W = { 0, 1, 2, 3, 4, 5, 6, 7….}
  • Set consisting of Integers: Z = { -2, -1, 0, 1, 2….}
  • Set consisting of Rational Numbers: Q = { ½, 2/3, ¾, -9/-5…} Here the elements are used in p/q form.
  • Set consisting of real numbers: R = { integers, whole numbers, rational numbers, irrational numbers }
  • Set consisting of positive numbers: S = { 1, 7, 10, 15, 21….}

Various types of sets in mathematics:

Sets can be differentiated into many types as related to mathematics. Let us check the following enlisted below.

  • Singleton sets: Singleton set is a type of set which has only a single element present within the set. For example, S = { x | x is an odd number between 6 and 8 }. Then S = { 7 }. So we can say that there is only one element in S. n (S) is therefore equal to 1.
  • Finite sets: Finite sets are those sets where the elements present are limited and can be countable. For example, A = { 20, 27, 34, 41, 48 }. Here within set A, we have five elements and thus it represents a finite set.
  • Infinite sets: Infinite sets can be defined as those sets where the elements present are unlimited and cannot be countable. For example, A = { 13, 15, 17, 19, 21, 23……}. Here within this set A, the total number of elements is not defined as it goes on and eventually cannot be counted. Hence, this kind of set can be termed an infinite set.
  • Empty sets: A set that does not contain any element is called an empty set. It is denoted by the Greek letter φ (phi). For example, A = { 6 < x <7 }. Here it says that the element would be greater than 6 but less than 7. Thus we can say that it is an empty set. Empty sets are often termed null sets or void sets.

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