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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
Symbol | Symbol Name | Meaning | Example |
{ } | set | The accumulation of elements | A = {1, 5, 9, 13, 15, 23}, B = {5, 13, 15, 21} |
A ∪ B | union | Elements that are a part of either set A or set B | A ∪ B = {1, 5, 9, 13, 15, 21, 23} |
A ∩ B | intersection | Those elements that are members of both sets, A and B | A ∩ B = {5, 13, 15 } |
A ⊆ B | subset | subset has a small number of or all of the elements equal to set | {5, 15} ⊆ {5, 13, 15, 21} |
A ⊄ B | not subset | members of the left set left set does not constitute a subset of the right set | {1, 23} ⊄ B |
A ⊂ B | proper subset / strict subset | subset has less elements compare to the set | {5, 13, 15} ⊂ {1, 5, 9, 13, 15, 23} |
A ⊃ B | proper superset / strict superset | set A has more elements compare to set B | {1, 5, 9, 13, 15, 23} ⊃ {5, 13, 15, } |
A ⊇ B | superset | set A has more or equal elements then set B | {1, 5, 9, 13, 15, 23} ⊃ {5, 13, 15, 21} |
Ø | empty set | Ø = { } | C = {Ø} |
P (C) | power set | all subsets of C | C = {4,5}, P(C) = {{}, {4}, {5}, {4,5}} Given by 2s, s is number of elements in set C |
A ⊅ B | not superset | set X is not a superset of set Y | {1, 2, 5} ⊅{1, 6} |
A = B | equality | both sets have the same members | {5, 13,15} = {5, 13, 15} |
Ac | complement | all the objects that are not in set A | We know, U = {1, 2, 5, 9, 13, 15, 21, 23, 28, 30} Ac = {2, 21, 28, 30} |
a∈B | element of | set membership | B = {5, 13, 15, 21}, 13 ∈ B |
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.
