Definition of Sets in Mathematics 

Sets, in math, is defined as a group of elements.

Concept of Set in Mathematics 

The concept of sets in mathematics deals with the properties and operations on collections of objects. This is particularly important for classification, organisation, and is the base for many forms of data analysis.

Three Methods of Describing Sets

There are mainly three methods of representing the elements within a set. They are enlisted below:

1.Statement Form: In the statement form, the accurate description and properties of a member of a set are written and enclosed within curly brackets. 

2.Roster Form: In the roster form of describing a set, each individual element of the set is listed within the curly brackets, with a comma for separation of elements. 

3.Set Builder Form: The set builder notation begins with an alphabet, say x, as a variable, followed by a colon. Then all the properties that an element x must satisfy in order to be considered a member of the set are then written. This notation is perfect to state all the properties of the elements of a particular set. 

Set Theory 

The essential features of the set theory include:

• Making of a set involves the grouping of objects of any kind into a single entity
• The specific relationship that may or may not exist between an object and a set is called a membership relationship. An object is a member of a set or it is not; there is no in-between. 
• The Principle of Extension states that the set is defined by its elements instead of a single means of defining the group. Thus, sets P and Q are equal only if all the elements they contain intersect, and none of them have any unique elements which are not present in the other set.

Sets and Relations Formulae 

The set theory formulas are listed below. For any three sets P, Q, and R:

• n ( P ∪ Q ) = n(P) + n(Q) – n ( P ∩ Q)
• If P ∩ Q = ∅, then n ( P ∪ Q ) = n(P) + n(Q)
• n( P – Q) + n( P ∩ Q ) = n(P)
• n( Q – P) + n( P ∩ Q ) = n(Q)
• n( P – Q) + n ( P ∩ Q) + n( Q – P) = n ( P ∪ Q )
• n ( P ∪ Q ∪ R ) = n(P) + n(Q) + n(R) – n ( P ∩ Q) – n ( Q ∩ R) – n ( R ∩ P) +  n ( P ∩ Q  ∩ R)

Properties of Sets

Commutative Property: When calculating the union or intersection of a set, changing the order of sets does not change the answer. Just like 2+8=8+2=10, even the following statements are true.

• P∪Q = Q∪P
• P∩Q = Q∩P

Associative Property: In an expression having two or more numbers or variables in a row of the same relational operators, the sequence in which the operations are performed does not make any difference in the result as long as the sequence of the operands is not changed. Just like (3+6)+4=(6+4)+3=13, the following statements also hold true.

• P ∪ ( Q ∪ R) = ( P ∪ Q) ∪ R
• P ∩ ( Q ∩ R) = ( P ∩ Q) ∩ R 

Distributive Property: Distributive law is when the operation is rearranged in a logical manner to arrive at the same results. For example, in arithmetic, 3x(2+8)=(3×2)+(3×8)=30, the same property is seen in set theory.

• P ∪ ( Q  ∩ R) = ( P ∪ Q)  ∩ (P ∪ R)
• P ∩ ( Q ∪ R) = ( P ∩ Q) ∪ ( P ∩ R) 

De Morgan’s Law: De Morgan’s Law holds that the complement of the intersection of the two sets is the union of their complements and the complement of the union of the two sets is the intersection of their complements. 

• Law of Union           : ( P ∪ Q )’ = P’ ∩ Q’
•  Law of Intersection : ( P ∩ Q )’ = P’ ∪ Q’

Complement Law: In set theory, the complement of set P refers to every element that is not present in set P. When all of the sets existing in the world are assumed to be subsets of a given set R, the absolute complement of A is the group of elements that is present in R but absent in A. 

• P ∪ P’ = P’ ∪ P =U
• P ∩ P’ = ∅ 

Idempotent Law and Law of Null and Universal Set: A idempotent element is an element, which when multiplied by itself, gives itself as the result. For example, 1 is idempotent is multiplication.

For any finite set P

• P ∪ P = P
• P ∩ P = P
• ∅’ = U
• ∅ = U’

Sets: Problems and Solutions

Problem: Represent a set of vowels in English in three ways.

• Statement form: {all vowels of the English alphabet}
• Roster Form: {a, e, i, o, u}
• Set Builder Form: {x: x is a vowel, x is a part of English alphabet}