Set Theory

Set Theory is an area of mathematical logic in which we study about sets and the properties of such sets, among other things. A collection of objects or groupings of objects is referred to as a set. These things are referred to as elements or members of a set in most cases. A set of players on a cricket team, for example, is a group of players.

Because a cricket team may only have a maximum of 11 players at any given moment, we can claim that this set is a finite set in this context. Another type of finite set is a collection of English vowels, which is another example of a finite set. Although many sets have infinite members (for example, a set of natural numbers, a set of whole numbers, a set of real numbers, a set of imaginary numbers, and so on), there are many sets that do not.

Representation of Sets

Roster Format

Using roster form, all of the items of the set are listed one after another, separated by commas, and encased between curly braces.

Consider the following example: If the set includes all of the leap years occurring between the years 1995 and 2015, it would be described in Roster form as: A ={1996,2000,2004,2008,2012}

Set Builder Form

In the set builder form, all of the elements have a common property that they all share. There are some items that are not part of the set that are not affected by this property.

S={ x: x is an even prime number}, where x is an even prime number, and ‘x’ is a symbolic representation used to express the element.

‘:’ is an abbreviation for ‘in such a way as’

‘The set of all’ is represented by the symbol ‘{}’.

As a result,S = { x:x is an even prime number }  can be understood as ‘the set of all x such that x is an even prime number’ in the context of prime numbers. S = 2 would be the roster form for this particular set S. Singleton/unit sets are what these types of sets are referred to as.

Types of Sets

 In basic set theory, there are several different sorts of sets. These are as follows:

• A finite set is one in which the number of elements is limited.

• empty set since it contains no elements.

• A singleton set is a set that has only one element.

• Equal set: Two sets are equal if they include the same elements in both of them.

• If the number of items in two sets is equal, then they are considered to be equivalent.

• A power set is a collection that contains every possible subset.

• Any set that contains all of the sets under discussion is referred to be a universal set.

• When all of the components in a set A are also elements of a set B, then A is a subset of B.

Mathematics Set Theory Symbols

Conclusion

Cantor came to the conclusion that the sets N and E are both equal in cardinality. Cantor went on to show that there is no one-to-one correlation between the set of real numbers and the set of natural numbers, which was previously assumed.