The Set Theory

A set is a mathematical model for a collection of objects; it contains elements or members, which can be any mathematical object: numbers, symbols, points in space, lines, other geometrical structures, variables, or even other sets. The empty set is a set with no elements, while a singleton is a set with only one element. If two sets have exactly the same elements, they are equivalent.

Basic About Set Theory:-

Set theory is an area of mathematics that teaches us about sets and their properties. A set can be defined as a  group of objects or a collection of objects. These items are frequently referred to as set elements or members. A set of cricket players, for example, is a group of players.

Because a cricket team can only have 11 players at any given moment, we can claim that this set is finite. A collection of English vowels is another example of a finite set. However, many sets, such as a set of natural numbers, a set of whole numbers, a set of real numbers, a set of imaginary numbers, and so on, have infinite members.

Origin of Set Theory:-

Georg Cantor (1845-1918), a German mathematician, was the first to propose the concept of “Set Theory.” He came across sets while working on “Problems on Trigonometric Series.” It will be impossible to explain other concepts like relations, functions, sequences, probability, geometry, and so on without first comprehending sets.

Definition of Set:-

A set is a well-defined collection of objects or people, as we learnt in the introduction. Many real-life instances of sets include the number of rivers in India, the number of colours in a rainbow, and so on.

Example:-

Consider the following example to better comprehend sets. While walking to school from home, Niva decided to jot down the names of nearby restaurants. The restaurants were listed in the following order:

List 1 : R₁           R₂            R₃              R₄               R₅

The list above is made up of many objects. It’s also well-defined. By well-defined, we mean that anyone should be able to determine whether or not an item belongs to a specific collection. A stationery store, for example, cannot be included in the restaurant category. A set is defined as a well-defined collection of items.

The elements of a set are the objects that make up the set. A set of elements can be finite or infinite. Niva wanted to double-check the list she had created earlier on her way home from school. She wrote the list in the order in which the eateries arrived this time. The revised list was as follows:

List 2 : R₅           R₄             R₃              R₂                R₁

This is an entirely distinct list. But if we consider the set, then it is the same as the set of list 1. In sets, the order of the elements has no bearing, hence the set remains the same.

Representation of Sets:-

There are two ways to express sets:

1. Roaster form

Example :- A = { a, o, i, u, e } 

                   S = { -1 , 1}

2. Set Builder form

Example :- A = { x : x is a vowel in English alphabet}

                   S = { x : x²-1 = 0}

Types of Sets:-

The sets are further divided into categories based on the components or types of elements they contain. Basic set theory distinguishes between the following sorts of sets:

• Finite Set: The set with a finite number of elements is called a finite set.
• Infinite Set: The set with an infinite number of elements is called an infinite set.
• Empty set: The set which has no elements is said to be an empty set.
• Singleton set: The set containing only one element is a singleton set.
• Equal Set: If two sets have the same elements, they are equal.
• Equivalent Set: If cardinality of two sets is equal, they are equivalent sets.
• Power Sets: A power set is a collection of all conceivable subsets.
• Universal Set: Any set that contains all the sets under discussion is referred to as a universal set.
• Subset: A is a subset of B when all of the elements of set A belong to set B.

Set Operations:-

The following are the four most commonly used set operations:

• Union of Sets ( ∪ )
• Intersection of Sets ( ∩ )
• Complement of Sets ( ‘ )
• Difference of sets ( – )

Set Theory Formulas:-

• n( A ∪ B ) = n(A) +n(B) – n (A ∩ B) 
• n(A∪B)=n(A)+n(B)   {when A and B are disjoint sets}
• n(U)=n(A)+n(B)–n(A∩B)+n((A∪B)’)
• n(A∪B)=n(A−B)+n(B−A)+n(A∩B) 
• n(A−B)=n(A ∪ B)−n(B) 
• n(A−B)=n(A)−n(A∩B) 
• n(A’)=n(U)−n(A)
• n(PUQUR)=n(P)+n(Q)+n(R)–n(P⋂Q)–n(Q⋂R)–n(R⋂P)+n(P⋂Q⋂R) 

Conclusion:-

Only set theoretic conceptions can precisely define many mathematical topics. Graphs, manifolds, rings, vector spaces, and relational algebras are all examples of mathematical structures that can be characterized as sets satisfying various (axiomatic) qualities. In mathematics, equivalence and order relations are common, and set theory can be used to describe the theory of mathematical relations.

The idea of introducing the principles of naïve set theory early in mathematics education has gained appeal as set theory has gained acceptance as a foundation for modern mathematics.