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Set theory is an area of mathematical logic that analyses sets, which are groupings of objects informally. Although any object can be gathered into a set, set theory as a discipline of mathematics is primarily concerned with those that are significant to all of mathematics.
Sets: A set can have any number of elements, and each object in the set is referred to as an element of the set. When writing a set, use this bracket { } , Set A = {1, 2, 3, 4, 5}. A set of items can be represented using a variety of notations.
Symbol of Sets
The elements of a set are represented by set symbols. The table below depicts some of these symbols and their meanings.
Symbol | Example |
{ } | {1, 2, 3, 4} |
A ∪ B | C ∪ D = {1, 2, 3, 4, 5} |
A ∩ B | C ∩ D = {3, 4} |
A ⊆ B | {3, 4, 5} ⊆ D |
A ⊂ B | {3, 5} ⊂ D |
A ⊄ B | {1, 6} ⊄ C |
A ⊇ B | {1, 2, 3} ⊇ {1, 2, 3} |
A ⊃ B | {1, 2, 3, 4} ⊃ {1, 2, 3} |
A ⊅ B | {1, 2, 6} ⊅ {1, 9} |
Formulas of sets
Formulas of sets given below:
n(A U B) = n(A) + n(B) – n(A ∩ B)
n (A ∩ B) = n(A) + n(B) – n(A U B)
n(A) = n(A U B) + n(A ∩ B) – n(B)
n(B) = n(A U B) + n(A ∩ B) – n(A)
n(A – B) = n(A U B) – n(B)
n(A – B) = n(A) – n(A ∩ B)
n(A U B) = n(A) + n(B)
A ∩ B = ∅
n(A – B) = n(A)
What is set theory with examples?
In all of mathematics, set theory is a fundamental notion. This branch of mathematics serves as a foundation for a variety of other subjects.
A set is, by definition, a collection of items known as elements. Although this appears to be a basic concept, it has far-reaching implications.
Elements: Numbers, states, cars, people, and even other sets are all options for elements in a set. Almost everything that can be collected together can be used to make a set, though there are a few factors to keep in mind.
Equal Sets: A set’s elements are either in the set or not. We can use a defining attribute to describe a set, or we can list the elements in the set. It makes no difference in the order they are listed. Because they both include the same elements, the sets {1, 2, 3} and {1, 3, 2} are equivalent.
Example: 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. Let’s utilize this example to learn the basics of arithmetic vocabulary related to sets.
How to do set theory
Set theory procedures are used to construct a set from two existing sets, just as we can conduct operations such as addition on two numbers to obtain a new number. There are other procedures, but nearly all of them are made up of the following three:
A union is a gathering together of two or more people. The components that are in either A or B make up the union of the sets A and B.
An intersection is a point where two things meet. The components that appear in both A and B make up the intersection of the sets A and B.
All of the items in the universal set that are not elements of A make up the complement of the set A.
Conclusion
In this article we learn that a set is a well-defined collection of things in mathematics. Sets are named and represented with a capital letter. The elements that make up a set in set theory can be anything: humans, letters of the alphabet, numbers, shapes, variables, and so on. Mathematicians use set theory all the time. Many subfields of mathematics rely on it as a foundation. It is extremely useful in the field of statistics, especially in the field of probability. Many of the notions in probability are drawn from set theory’s consequences. Set theory is one means of expressing the principles of probability.
