Types of Set

# Types of Set | Algebra for IIT JAM Mathematics PDF Download

## Definition - Set theory

The set is a well-defined collection of definite objects of perception or thought.
• A set may also be thought of as grouping together of single objects into a whole.
• The objects should be distinct from each other and they should be distinguished from all those objects that do not form the set under consideration. Hence a set may be a bunch of grapes, a tea set or it may consist of geometrical points or straight lines.
• A set is defined as an unordered collection of distinct elements of the same type where the type is defined by the writer of the set.
• Generally, a set is denoted by a capital symbol and the master or elements of a set are separated by commas and enclosed in { }. More symbols used in the set are given below:
1 ∈ A means 1 belongs to set A
1 ∉ A  means 1 does not belong to A

## Types of Set

There are many types of set in the set theory:

### 1. Singleton Set

If a set contains of only one element it is called to be a singleton set.

Examples:

• The sets given by  {1}, {0}, {a} are all consisting of only one element and therefore are singleton sets.
• E = {x : x ϵ N and x3 = 27} is a singleton set with a single element {3}
• W = {v: v is a vowel letter and v is the first alphabet of English} is also a singleton set with just one element {a}.

### 2. Finite Set

As the name implies, a set with a finite or exact countable number of elements is called a finite set. If the set is non-empty, it is called a non-empty finite set.

Examples:

• A = {a, e, i, o, u} is a finite set because it represents the vowel letters in the English alphabetical series.
• B = {x : x is a number appearing on a dice roll} is also a finite set because it contains – {1, 2, 3, 4, 5, 6} elements.

### 3. Infinite Set

A set with an infinite number of elements is called an infinite set. In other words, if a given set is not finite, then it will be an infinite set.

Examples:

• C = {p: p is a prime number} is an infinite set.
• D = {k: k is a real number} is also an infinite set.

Finite and Infinite Sets

### 4. Equal and Unequal Sets

Two sets X and Y are said to be equal if they have exactly the same elements (irrespective of the order of appearance in the set). Equal sets are represented as X = Y. Otherwise, the sets are referred to as unequal sets, which are represented as X ≠ Y.

• If X = {a, e, i, o, u} and H = {o, u, i, a, e} then both of these sets are equal.
• If C = {1, 3, 5, 7} and D = {1, 3, 5, 9} then both of these sets are unequal.
• If A = {b, o, y} and B = {b, o, b, y, y} then also A = B because both contain same elements.

Question for Types of Set
Try yourself:{x: x is a real number between 1 and 2} is an ______

### 5. Null Set/ empty Set

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or φ (called phi)

Examples:

• Let A = {x : 9 < x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10. Therefore, A = {} or φ
• Let W = {d: d > 8, d is the number of days in a week} will also be a void set because there are only 7 days in a week.

### 6. Subset and Super Set

A set S is said to be a subset of set T if the elements of set S belong to set T, or you can say each element of set S is present in set T. Subset of a set is denoted by the symbol (⊂) and written as S ⊂ T and  T is the superset of set S(T ⊇ S).

Example:

Example: A = {1,2,3} B = {1,2,3,4,5,6}
B, since all the elements in set A are present in set B.
B ⊇ A denotes that set B is the superset of set A.

In the given Venn diagram, A is the subset of set B.

### 7. Equivalent Sets

Equivalent sets are those which have an equal number of elements irrespective of what the elements are .

Examples:

• A = {1, 2, 3, 4, 5} and B = {x : x is a vowel letter} are equivalent sets because both these sets have 5 elements each.
• S = {12, 22, 32, 42, …} and T = {y : y2 ϵ Natural number} are also equal sets.

Question for Types of Set
Try yourself:Subset of the set A = {} is?

### 8. Disjoint Sets

Two sets are disjoint sets if there are no common elements in both sets.

Example:

A = {1,2,3,4} B = {7,8,9,10}. Here, set A and set B are disjoint sets.

### 9. Power Set

Power set of a set is defined as a set of every possible subset. It is denoted by P(A).

If the cardinality of A is n then the Cardinality of the power set is 2n as every element has two options either to belong to a subset or not.

Example:

• Set A = {1,2,3}. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

### 10. Universal Set

Any set which is a superset of all the sets under consideration is said to be universal set and is generally denoted U.

The universal set is represented by the letter U.

Example:
Let  A = {1, 2, 3}

C = { 0, 1} then we can take

S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.

## Solved Examples for You

Question 1: If A = {x: x is an even natural number} and B = {y: y is the outcome of a dice roll}, determine the nature of the two sets.
Answer : A = {2, 4, 6, 8, 10, 12, 14, …} And B = {1, 2, 3, 4, 5, 6}. So, set A is an infinite set while set B is a finite set.

Question 2: If X = {1, 2, 3, 4, 5}, Y = {a, e, i, o, u} and Z = {u, o, a, i, e}; determine the nature of sets.
Answer: Since the pairs of sets X – Y, Y – Z as well as Z – X have the same number of elements, i.e. 5 they are EQUIVALENT sets. And sets Y and Z are also EQUAL sets because apart from having the number of elements the same, they also have the same elements, i.e. the alphabets of English vowel letters.

Question 3: What is the classification of sets in mathematics?
Answer: There are various kinds of sets like – finite and infinite sets, equal and equivalent sets, a null set. Further, there are a subset, power set, universal set in addition to the disjoint sets with the help of examples.

Question 4: What are the properties of sets?
Answer: The fundamental properties are that a set can consist of elements and that two sets are equal, if and only if every element of each set is an element of the other; this property is referred to as the extensionality of sets.

Question 5: If A = {R,O,Y,A,L} and B = {L,O,Y,A,L}, determine the type of the two sets to be equal or unequal.

Answer: A = {R,O,Y,A,L} and B = {L,O,Y,A,L}. So, sets A and B are unequal sets as the elements of set A are not similar to the elements of set B.

The document Types of Set | Algebra for IIT JAM Mathematics is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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## FAQs on Types of Set - Algebra for IIT JAM Mathematics

 1. What is set theory?
Ans. Set theory is a branch of mathematics that deals with the study of sets, which are collections of well-defined objects. It provides a foundation for various mathematical disciplines and is used to analyze relationships between sets and their elements.
 2. What are the different types of sets?
Ans. There are various types of sets in set theory, including: - Finite sets: Sets with a finite number of elements. - Infinite sets: Sets with an infinite number of elements. - Empty set: A set with no elements. - Singleton set: A set with exactly one element. - Universal set: A set that contains all possible elements under consideration.
 3. Can a set have duplicate elements?
Ans. No, by definition, a set cannot have duplicate elements. Each element in a set is unique, and repetition is not allowed. If an element appears more than once in a collection, it is considered as a single element in the set.
 4. How are sets represented in set theory?
Ans. Sets are usually represented by listing their elements within curly braces. For example, the set of prime numbers less than 10 can be represented as {2, 3, 5, 7}. Alternatively, set builder notation can be used to define sets based on specific properties or conditions.
 5. What is the importance of set theory in mathematics?
Ans. Set theory is fundamental to mathematics as it provides a framework for defining and studying mathematical objects. It helps in formalizing mathematical concepts, reasoning about relationships between objects, and providing a foundation for other branches of mathematics, such as algebra, calculus, and logic. Additionally, set theory plays a crucial role in the development of mathematical logic and proofs.

## Algebra for IIT JAM Mathematics

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