Sets Formula

Basic Concepts of Set Theory

Set is a group of clearly distinguishable objects which are well-marked from each other. Sets in mathematics are generally denoted by capital letters A, B,C,… and elements are generally denoted by small letters a, b, c,… A set consisting of definite elements is called finite sets; else, it is an infinite set. Few Key Points to Remember is that for any set A, every set is a subset of itself i.e., A ⊆ A. Also, for any set A, an Empty set Φ is a subset of every set i.e., Φ ⊂ A.


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Sets Theory Formulas

These are standard formulas in mathematics from the set theory. This is to say, If there are two sets namely A and B, then

  1. n(A U B) depicts the number of elements that exist in one of the sets A or B.

  2. n(A ⋂ B) depicts the number of elements that exists in both the sets A and B.

  3. n(A U B) = n(A) + (n(B) – n (A ⋂ B).

For three sets A, B, and C

  • n(AUBUC)=n(A)+n(B)+n(C)–n(A⋂B)–n(B⋂C)–n(C⋂A)+n(A⋂B⋂C)

Subset

A set P is said to be a subset of set Q if each element of set P belongs to set B. Symbolically, we write it as

P ⊆ Q, if x ∈ P ⇒ x ∈ Q

Find the important symbols for Sets below:


Set Notations

Symbol

Set

N

A set of all the natural numbers

R

A set of all the real numbers

R+

A set of all the positive real numbers

Q

A set of all the rational numbers

Z

A set of all the integers

Z+

A set of all the positive numbers


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Use of Venn-Diagrams in Set Theory

Venn diagrams are the mathematical diagrams, which are drawn to illustrate a clear connection between sets. In a Venn-diagram, the universal set U is described by a point enclosed in a rectangle while its subsets are described by points within the closed curves (generally circles) inside the rectangle.


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Solved Examples for Sets Formula

Example1:

In a group of 100 sports players, 25 prefer playing basketball and 40 prefer playing volleyball. 10 like both. Find out how many like either of them and how many like neither?


Solution1:

Given that the Total number of players, n(µ) = 100

Number of basketball players, n(S) = 25

Number of volleyball players, n (M) = 40

Number of players who prefer both, n (M∩S) = 10

Number of players who prefer either of them, n {MᴜS} = n {M} + n{S} – n {M∩S}

→ 25 + 40 -10 = 55

Number of students who like neither = n (µ) – n (MᴜS) = 100 – 55 = 45


Thus we get our answer

  1. Those who like either of them = 55

  2. Those who like neither of them = 45

The simplest way to solve set equations problems is by drawing Venn diagrams, as shown below. This is the final Venn diagram after obtaining the solution to the set problem.


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Example 2:

There are 50 students in a dance class. Among them, 10 students are learning both salsa and hip-hop. A total of 25 students are learning salsa. If every student is learning at least one dance form, how many students are learning Hip-hop in total?


Solution 2:

Given that, every student is learning at least one dance form. Thus, there are no students that belong to the category ‘neither’.

So in this condition, n(EᴜF) = n(µ).

Further, it is given in the problem that a total of 25 students are learning salsa. This clearly DOES NOT mean that 25 are learning ONLY Salsa. Only when the term ‘only’ is mentioned in the problem should we consider it so, otherwise not.


Now that, 25 are learning salsa and 10 are learning both. This means that 25 – 10 = 15 are learning ONLY Salsa.

S0, n (µ) = 50, n (E) = 15

N (EᴜF) = n(E) + n(F) – n(E∩F)

50 = 25+ n(F) – 10

N (F) = 35

Hence, total number of students learning Hip-hop = 35

Note: The question is only about the total number of students learning Hip-hop and not about those learning ONLY hip-hop, which would have been a different answer, i.e. 25.


Fun Facts

  1. Equal sets are invariably equivalent but equivalent sets may not always be equal.

  2. The set containing {Φ} is not a null set, in fact is a set containing one element Φ.

FAQ (Frequently Asked Questions)

1. Is there any Technique to Define a Set?

Ans: Yes, we have different Methods for Describing a Set that are as follows:

  1. Tabular Form: Also known as Roster or Listing Method, this method describes a set by listing elements, which is separated by commas, within braces. For example: A = [v, w, x, y, z].

  2. Set Builder: Also commonly known as the Rule Method, it describes a set by writing down a property or rule which provides us with all the elements of the set by that property. For example: A = {u : u is a vowel of English letters}.

2. Are there any Categories of Sets?

Ans: Sets are classified based on their elements and properties. There are different types of sets which are as below:

  1. Empty Sets: A set which does not consist of any element is known as an empty set or null set or the void set. It is denoted by {} or Φ.

  2. Equal Sets: Two sets A and B are said to be equal, in case each element of A is also an element of B or vice-versa, i.e. two equal sets will contain exactly the same element.

  3. Finite and Infinite Set: A set that contains a finite number of elements, is known as a finite set, else the set is called an infinite set.

  4. Equivalent Sets: Two finite sets A and B are called equal if the number of elements are equivalent to each other, i.e. n(A) = n(B)

  5. Singleton Set: A set which contains a single element, is called a singleton set.