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A Set is the collection of objects called elements of the set. When we talk about sets, it is the most elementary building block used in mathematics. Once sets are introduced one can compare them, define operations similar to addition and multiplication on them and they can be used to define new objects such as various kinds of number systems and the elements belonging to different groups of Numbers. Most of the topics in modern analysis are based on the sets operations. We can say that sets are the collection of the objects selected from some universe called the universal set, which is largest among all sets. Sets are denoted by capital, bold letters or curly brackets. We can represent the sets by writing the elements of the sets in the curly braces separated by commas.

When we study Probability and sets together, then the probability of occurrence of any event from the given set forms probability sets. Probability set can be seen in the following example: Let’s have a set of Prime Numbers between 1 to 20. The set so formed will be P = 2, 3, 5, 7, 11, 13, 17. Now, we need to find the set probability of getting the product of any three randomly chosen numbers less than 31. We observe that only 3 numbers 2, 3, and 5 belong to this set as the product 2 * 3 * 5 = 30, which is less than 31. So we say the probability will be 3 / 7, where 7 is the total number of elements in the set of prime numbers below 20 and three numbers are there whose product is less than 31. As there are 3 numbers, so the number of ways these numbers can be arranged are factorial 3 = 3!= 3 * 2 * 1 = 6 ways.

Subsets

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A Set is a well defined collection of the object, where each object is called an element of the set. A set is always represented in capital letter like A, B, C, etc. The elements in the set are usually represented by small letters like x, y, z. So we say let set A contains the element 2, 4, 6, 8, 10. So it is written as A= 2, 4, 6, 8, 10. The elements of any set are represented inside the curly braces. Any given two Sets are said to be equal, if they have exactly same number of elements in it. We say a set is a null set or an empty set, if the number of elements in the set is 0. Now we will learn about Probability subsets. Any set A is called a subset of set B, if each element of set A is contained in set B. It is represented by a symbol called belongs to. We can write A ⊂ B.

Sometimes we describe a set with the help of a statement. Example: Set A is a set of all even Numbers less than 10 and a set B is a set of Natural Numbers less than equal to 10. Here we observe that all the elements of set A belong to the elements of set B. So we can say set A is the subset of set B. We may say: A = 2, 4, 6, 8.

A set A is a subset of a set B or we can also say that set B is superset of set A. It also means that all the elements of set A are present in set B. Sometimes we come across the situations when subset A and superset B coincide. Which means all the elements of set A are same as the elements of set B. Order of both the sets is also same, which means that the number of elements in both the sets are equal.

Power Sets

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To understand power Sets, let us take a Set A = a, b, c. We observe that this set ‘A’ has 3 elements, so three Subsets of the set ‘A’ can be a, b, and c. Moreover we can say that a set ‘A’ can also have a subset a, b, c which is called a proper subset of set ‘A’. Another important thing we must remember is null set, which is an empty subset of a set ‘A’. We conclude that if we prepare a list of all the subsets of set ‘A’, it will form a power set of set ‘A’. A power set is represented by ‘P (A)’. The order of a power set can be calculated, if we know the number of elements of the set ‘A’. If there are ‘n’ elements in set ‘A’, then the order of Power set is 2n. They are called the members of power set. So, as in above set ‘A’, we see that there are 3 elements a, b, and c, so the order of its power set is 23 = 2 * 2 * 2 = 8.
A power set can also be represented by |S| or it can also be written as | P(S)| = 2n. Now a question arises that, why we use 2 with a power ‘n’ to represent the order of power set. It is also called a binary set ‘s’, the formation of a power set is based on the formation of Binary Numbers. If we write the sequence of binary numbers of all ‘n’ digits, where 1 means to represent the matching element of the digit. We can understand the concept with the help of table shown below-

Number a b c Subset
0 0 0 0 Null set
1 0 0 1 c
2 0 1 0 b
3 0 1 1 b, c
4 1 0 0 a
5 1 0 1 a , c
6 1 1 0 a, b
7 1 1 1 a, b, c

Cardinality

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Any Set ‘S’ which has ‘n’ different elements for some natural number ‘n’, then ‘n’ is called the cardinality of set ‘S’. It is also called as size of set ‘S’ and here ‘S’ is a finite set. If we have any set S = 1, 2, 3 in this set we have 3 elements, so the cardinality of set ‘S’ is 3. Word cardinality of Sets means the basic members in the family. Here the family of set containing the number of members i.e., the elements is called cardinality of sets. A cardinality set can be finite, i.e. any positive Integer or sometimes it can be infinite too.
If we have any set of colors say B= RED, BLUE, GREEN, YELLOW, WHITE, here we find that the number of members in the set of colors is 5 so we say |B| = 5 or we can say the cardinality of set ‘B’ is 5. Any set with no members is called a null set. The cardinality of a null set is zero, as the number of elements in this set is zero. If we write B = 0 , it represents a null set and it is different from set A= φ . Here we find the cardinality of set ‘B’ is zero, on other hand the cardinality of set ‘A’ is 1 as there is 1 element in this set.
Now for instance we take a set of natural Numbers, we know Natural Numbers are infinite, so cardinality of the set ‘N’ of natural numbers will also be infinite as there are infinite numbers of elements in this set. If we take a set of Real Numbers, we know that real numbers are also not countable and are infinite in number, so the cardinality of set of real numbers is also infinite.

Special Sets

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We say that the Sets of numbers which are used as the special names and symbols are called special sets. Some of the special sets are as follows:
1. A Set of Natural Numbers: The set consisting of series of numbers 1, 2, 3, …..up to infinity are called as special set of Natural numbers. It is represented by ‘N’.
2. A set of whole numbers: The set consisting of series of numbers 0, 1, 2, 3, …..up to infinity are called as special set of Whole numbers. It is represented by ‘W’
3. A set of integers: The set consisting of series of all Positive and Negative Numbers including zero …. -3, -2, -1, 0, 1, 2, 3, ….. up to infinity are called as special set of Integer numbers. It is represented by ‘Z’
4. A set of Rational Numbers: When a set consisting of the numbers expressed in form of p/q, where ‘p’ and ‘q’ both are integers and q <> 0, then such set of numbers is called the set of rational numbers and it is expressed by ‘Q’. All decimal, fractions and ordinary numbers belong to the family of rational numbers.
5. A set of Irrational Numbers: The numbers which cannot be expressed in form of p/q are called irrational numbers. They are mostly non repeating, non Terminating Decimal numbers and Square roots and Cube roots of numbers
6. A set of complex Numbers: A set of numbers with real and imaginary part written together are called set of complex numbers.
Similarly a set of imaginary numbers is expressed by ‘I’; Algebraic numbers is expressed by ‘A’. We also observe that a set of natural numbers is a subset of Integers, Integers are the Subsets of Rational numbers and the set of Rational numbers are the subset of Real Numbers.

Types of Sets

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A collection of different objects or we can say entities is known as Sets but these objects must be distinct and well defined. Sets is one of the most important concepts of mathematics from which different concepts of mathematics have been derived. The objects or entities which are included in a Set are also called elements of that set. Sets are basically denoted by capital letters. Two Sets are called as equal sets if elements contained by both sets are same. Example of set can be collection of positive Numbers which can be expressed as A = {1, 2, 3, 4}. There are different types of sets which are discussed below.

Different kinds of sets are:

Empty Set: A set with no element in it is called as Empty Set. It is also called as Null set or Void Set.

Singleton Set: A set that contains only one element is known as singleton set. Like Set X = {8} is singleton set.

Finite and Infinite set: A set with finite number of elements whose number is known is known as finite set and a set with infinite number of elements is called as infinite set.

Disjoint of sets: Two sets are called disjoint sets when there is no common element among them.

For example: A ={5, 6, 7} and B = {2, 3}, these sets have no common elements so these sets are disjoint sets.

Equal Sets: Two sets are 'A' and 'B' are called equal sets if A ⊆ B and B ⊆ A, so A = B.

Example of equal sets: A = {5, 6, 7} and B = {5, 6, 7} therefore A = B.

These is all about types of sets.

Law of Sets in Math

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Sets in mathematics can be defined as a collection of well defined and distinct objects. The entities or objects contained by a Set are known as elements of set. We can represent Sets using capital letters. Example of Sets is A = {1, 2, 3, 4} where 'A' is the name of set and 1, 2, 3, and 4 are elements of set A. Now we will study Law of sets in Math.
There are different laws in sets Algebra which are discussed below:
Commutative laws: If we have two sets 'P' and 'Q' then according to commutative law:
P ∪ Q = Q ∪ P and P ∩ Q = Q ∩ P.
Associative Laws: If we have three sets 'P', 'Q' and 'R' then according to associative law:
(P ∪ Q) ∪ R = P ∪ (Q ∪ R) and (P ∩ Q) ∩ R = P ∩ (Q ∩ R).
Distributive Laws: If we have three sets 'P', 'Q' and 'R' then according to distributive law:
P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R) and P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R).
Identity Laws: If we have a set 'P' then according to identity law: P ∪ ∅ = P and P ∩ U = P, where '∅' is an empty set and 'U' is universal set.
Complement Laws: If we have a set 'P' then according to complement law:
P ∪ P c = U and P ∩ P c = ∅, where P c is complement of set P.
This is all about Law of sets in Math.