Introduction to Sets

In the early twentieth century, John-Euler Venn solved the word problems in arithmetic with the help of Venn-diagram. This method grew popular as it is easy to understand and simple to calculate. So, Venn-diagram is associated with his name.

Types of sets

Empty/Null set

A set which does not contain any element is called an empty set or null set. It is denoted by \(\emptyset\)
empty, and is read as phi. In roster form, \(\emptyset\) it is denoted by { }.
For example:

  1. The set of the whole number less than zero. Clearly there is no whole number less than zero.
    Therefore, it is an empty set.
  2. Let A = {x:2 < x <3, x is a natural number}
    Here A is an empty set because there is no natural number between 2 and 3.


\(\emptyset\) ≠ { 0 }

{ 0 } is a set having one element.

Singleton set

A set which has only one element is called singleton set.
For example:

  1. A = { x:x neither prime nor composite}
    It is a singleton set only containing one element i.e. 1.
  2. Let B = { x:x \(\in\) N and x2 = 4}
    It is a singleton set only containing one element i.e. 2, which square is 4.

Finite set

A set which contains a definite number of elements is called a finite set. The empty set is also called a finite set.
For example:

  1. The set of all color in the rainbow.
  2. A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, ........................................ 50 }

Infinite set

A set which contains never ending element is called infinite set.
For example:

  1. Set of all point in the plain.
  2. Set of all prime number.
  3. B = { x:x \(\in\) W, x = 2n }

Cardinal number of set

The number of distinct element in a given set A is called the cardinal number of A. It is denoted by n(A).
For example:

  • B = { x:x \(\in\) N, x<5}
    B = {1,2, 3, 4}

Equivalent set

Two set are called to be equivalent if their cardinal number is same i.e. n(A)=n(B). The symbol for denoting an equivalent set is '\(\leftrightarrow\)'.
For example:

A = {1, 2, 3, 4, 5}
B={a, b, c, d, e }
Therefore, A \(\leftrightarrow\) B.

Equal set

Two set are said to be equal if they contain the same element. Every element of a set A has every element of B and every element of B has every element of A.
For example:

A= { a,e, i,o, u }
B= {u, i, o, a, e}

\(\therefore\) A = B

  1. n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(C∩A) - n(B∩C) + n(ABC)
  2. Maximum value of n(A∩B) ↔ minimum value of n(A∩B)
  3. Minimum value of n(A∩B) ↔ Maximum value of n(A∩B)
  4. The cardinality of a set is a positive integer but it is not decimal. So n(A)  ≠ 50% because 50% = 0.5
  5. Total of cardinality shown in Venn-diagram must be equal to the cardinality of ∪.



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Sign of intersection


if 7 is sabtraced from haif a sqare number then the result is 25 find the numers