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Type Of Sets

  • Last Updated : 26 May, 2021

Things look better when they are in a well-arranged order. In mathematics, numbers, objects, or anything that can be grouped or can be shown collectively are arranged and are defined as Sets. Sets are important to learn not only to understand the concepts in math but to also apply them in day-to-day life since arranging things that belong to the same category and keeping them in one group helps to locate things easily and looks neat as well. One example is keeping out clothes in that manner, say, the socks will go in one set, and the shirts will go in another set, making such sets and keeping the items in their respective groups will now help in easily finding any clothing item. This is just one real-life example of how sets work. Let’s learn about sets in detail.

Sets

A well-defined collection of Objects or items or data is known as a set. The objects or data are known as the element. For Example, the boys in a classroom can be put in one set, all integers from 1 to 100 can become one set, all prime numbers can be called an Infinite set. The symbol used for sets is {…..}. What looks like a set but can not be called a Set? Basically, Sets do not do favoritism, therefore, things that include any quality or characteristics of objects, are not to be put in a Set. For Instance, “All the Students having poor handwriting” can not be placed in a set.

Question: Separate out the collections that can be placed in a set.

  1. The beautiful Girls in the Park
  2. All even numbers
  3. All the good basketball players
  4. The natural numbers divisible by 3
  5. Number from 1 to 10.

Answer:

Anything that tries to define a certain quality or characteristics can not be put in a set. Hence, from the above given Collection of data. 



The ones that can be a set,

2. All even numbers

4. The natural numbers divisible by 3.

5. Number from 1 to 10

The ones that cannot be a set,

1. The beautiful girls in the park

3. All the good basketball players.

Types of Sets

Sets are the collecting of different elements belonging to the same category and there can be different types of sets seen. A set may have an infinite number of elements, may have no elements at all, may have some elements, may have just one element, and so on. Based on all these different ways, sets are classified into different types.



The different type of sets are:

Singleton Set

Singleton Sets are those sets that have only 1 element present in them. For instance, Set A= {1} is a singleton set as it has only element, that is, 1. Set B= {Reo} is also a singleton set as there is only one element which is a name. Similarly, all the sets that contain only one element are known as Singleton sets.

Empty Sets

Empty sets are also known as Null sets or Void sets. They are the sets with no element/elements in them. They are denoted as ϕ. Examples of Null sets are,

  • Set A= {a: a is a number greater than 5 and less than 3}
  • Set B= {p: p are the students studying in class 7 and class 8}

Finite Sets 

Finite Sets are those which have a finite number of elements present, no matter how much they’re increasing number, as long as they are finite in nature, They will be called a Finite set.

Example: 

1. Set A= {a: a is the whole number less than 20}

2. Set B = {a, b, c, d, e}



Infinite Sets

Infinite Sets are those that have an infinite number of elements present, cases in which the number of elements is hard to determine are known as infinite sets. 

Example: 

1. Set A= {a: a is an odd number}

2. Set B = {2,4,6,8,10,12,14,…..}

Equal Sets

Two sets having the same elements and an equal number of elements are called equal sets. The elements in the set may be rearranged, or they may be repeated, but they will still be equal sets.

Example: Set A = {1,2,6,5}

Set B = {2, 1, 5, 6}

In the above example, the elements are 1, 2, 5, 6. Therefore, A= B.

Equivalent Sets

Equivalent Sets are those which have the same number of elements present in it. It is important to note that the elements may be different in both sets but the number of elements present is equal. For Instance, if a set has 6 elements in it, and the other set also has 6 elements present, they are equivalent sets.

Example: Set A= {2, 3, 5, 7, 11}

Set B = {p, q, r, s, t}

Set A and Set B are equivalent sets.

Subsets

Set A will be called the Subset of Set B if all the elements present in Set A already belong to Set B. The symbol used for the subset is ⊆ 

If A is a Subset of B, It will be written as A ⊆ B

Example: Set A= {33, 66, 99}, Set B = {22, 11, 33, 99, 66}

Then, Set A ⊆ Set B 



Universal Sets 

A universal set is a set that contains all the elements of the rest of the sets. It can be said that all the sets are the subsets of Universal sets. The universal set is denoted as U.

Example: Set A ={a, b, c, d}

Set B= {1,2}

U= {a, b, c, d, e, 1, 2}

Sample Problems

Question 1: What is the difference between, ϕ and {ϕ}.

Answer:

  • ϕ = this symbol is used to represent the null set, therefore, when only this symbol is given, the set is a Null set or empty set.
  • {ϕ}= In this case, the symbol is present inside the brackets used to denote a set and therefore, now the symbol is acting like an element. Hence, this is a Singleton set.

Question 2: Represent universal set on a Venn Diagram.

Answer:

Universal Sets are those that contain all the sets in it. In the below given Venn diagram, Set A and B are given as examples for better understanding of Venn Diagram.

Example: Set A= {1,2,3,4,5}, Set B = {1,2, 5, 0}

U= {0, 1, 2, 3, 4, 5, 6, 7}

Question 3: Which of the given below sets are equal and which are equivalent in nature?

  1. Set A= {2, 4, 6, 8, 10}
  2. Set B= {a, b, c, d, e}
  3. Set C= {c: c ∈ N, c is an even number, c ≤ 10}
  4. Set D = {1, 2, 5, 10}
  5. Set E= {x, y, z}

Answer:

Equivalent sets are those which have the equal number of elements, whereas, Equal sets are those which have the equal number of elements present as well as the elements are same in the set.

Equivalent Sets ⇢ Set A, Set B, Set C.

Equal Sets ⇢ Set A, Set C.

Question 4: Determine the types of the below given sets,

  1.  Set A= {a: a is the number divisible by 10}
  2. Set B = {2, 4, 6}
  3. Set C = {p}
  4. Set D= {n, m, o, p}
  5. Set E= ϕ

Answer:

From the knowledge gained above in the article, the above-mentioned sets can easily be identified.

  1. Set A is an Infinite set.
  2. Set B is a Finite set
  3. Set C is a singleton set
  4. Set D is a Finite set
  5. Set E is a Null set

Question 5: Explain which of the following sets are the subsets of Set P,

Set P = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

  1. Set A = {a, 1, 0, 2}
  2. Set B ={0, 2, 4}
  3. Set C = {1, 4, 6, 10}
  4. Set D = {2, 20}
  5. Set E ={18, 16, 2, 10}

Answer:

  1. Set A has elements a, 1, which are not present in the Set P. Therefore, set A is not a Subset.
  2. Set B has elements which are present in set P, Therefore, Set B ⊆ Set P
  3. Set C has 1 as an extra element. Hence, not a subset of P
  4. Set D has 2, 20 as element. Therefore, Set D ⊆ Set P
  5. Set E has all its elements matching the elements of set P. Hence, Set E ⊆ Set P.

Attention reader! Don’t stop learning now. Join the First-Step-to-DSA Course for Class 9 to 12 students , specifically designed to introduce data structures and algorithms to the class 9 to 12 students




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