Sets Formula

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

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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.