# Different Types of Sets

1. A set having finite number of elements is called a finite set.
2. A set which is not a finite set is called an infinite set.

For example, let A = set of all points on a particular straight line. Here A is an infinite set.
3. A set having no element is called a null set or an empty set or a void set. It is denoted by Ï† or {}.
4. A set having single element is called a singleton set, e.g., {Ï†}, {1}, {3}.
5. Two finite sets A and B are said to be equivalent if they have the same cardinal number. Thus sets A and B are equivalent iff n(A) = n(B).
6. Two sets A and B are said to be equal if each element of A is an element of B and each element of B is an element of A.
Notes:
• Two sets A and B are equal if x âˆˆ A â‡’ x âˆˆ B and x âˆˆ B â‡’ x âˆˆ A.
• Equal sets are equivalent sets but equivalent sets may or may not be equal.
1. Subsets: A set A is said to be a subset of a set B if each element of A is also an element of B. If A is a subset of set B, we write A âŠ† B. Thus, A âŠ† B â‡” x âˆˆ A â‡’ x âˆˆ B.
• Every set is its own subset.
• Empty set is a subset of each set.
• Let A and B be any two sets, then A = B â‡” A âŠ† B and B âŠ† A.
• Let A, B, C be three sets. If A âŠ† B and B âŠ† C, then A âŠ† C.
2. A set A is said to be a proper subset of a set B if A is a subset of B and A â‰  B.
3. A set A is said to be a superset of set B, if B is a subset of A, i.e., each element of B is an element of A. If A is a superset of B, we write A âŠ‡ B.
4. The set or family of all the subsets of a given set A is said to be the power set of A and is denoted by P(A). Symbolically, P(A) = {X : X âŠ† A}. Thus, X âˆˆ P(A) â‡” X âŠ† A. If A has n elements then its power set P(A) has 2n elements, i.e., A âŠ† B â‡’ P(A) âŠ† P(B).
5. In any discussion in set theory we need a set such that all sets under consideration in that discussion are its subsets. Such a set is called the universal set for that discussion.