Types of Sets
Sets can be classified into many categories. Some of which are finite, infinite, subset, universal, proper, power, empty etc.
Finite Sets
A set is said to be finite if it contains specific or finite number of elements. Finite sets are also called numerable or countable sets.
Countable finite set
A set is called a countable finite set if there is one-to-one correspondence between the elements in the set and elements in the set N, where N is the set of natural numbers.
Example: The set of alphabets in English Language.
As the number of all the elements of the given set is fixed i.e. 26 elements and one-to-one mapping of all the elements is also possible. Hence it is a countable finite set.
Infinite Sets
A set which is not finite or a set that contains infinite elements is called as an infinite set. An infinite set can be countable set or uncountable set.
Countable infinite set
If there is one-to-one correspondence between the elements in the set and elements in the set N, where N is the set of natural numbers. A countably infinite set is also known as Denumerable. A set that is either finite or denumerable is known as countable.
Example: The set of a non-negative even integer.
As the set of non-negative even integer is infinite, and the set of natural numbers is also infinite, but every element of set of non-negative even integer forms one-to-one unique mapping. Hence it is a countable infinite set.
Uncountable infinite set
A set which is not countable is called uncountable infinite set or non-denumerable set. If there is no one-to-one correspondence between between elements of the set and the set of natural numbers.
Example: The Set of all positive even numbers.
As there are infinitively many real numbers and between any two real numbers, there are infinite numbers. So the one-to-one mapping of all the elements with the set of natural numbers seems to be impossible. Hence it is an uncountable infinite set.
Subsets
If every element in a set A is also an element of a set B, then A is called a subset of B and B is called superset of A. It can be denoted as A ⊆ B.
Example: If A = {1, 2, 3} and B = {1, 3, 4, 2, 5}, then A is the subset of B or A ⊆ B.
If A is not a subset of B, i,e., if at least one element of A does not belong to B, we write A ⊈ B or B ⊈ A.
- Every set is a subset of itself, i.e. A ⊆ A.
- If A is a subset of B and B is a subset of C, then A will also be a subset of C. If A ⊆ B and B ⊆ C ⇒ A ⊆ C.
- A finite set having n elements has 2n subsets.
Proper Subset
If A is a subset of B and A ≠ B, then A is said to be a proper subset of B. It is denoted as A ⊂ B.
Example: If A = {1, 2, 3} and B = {1, 3, 4, 2, 5}, then A is the proper subset of B or A ⊂ B.
Note: A can never be a proper subset of itself i.e. A ⊄ A.
If A is proper subset of B, then B can never be the subset of A.
If A ⊆ B and A ≠ B ⇒ A ⊂ B ⇒ B ⊈ A.
Null Set or Empty Set
A unique set having no elements is called a null set or void set. It is denoted by ∅ or {}.
Note: |∅| = 0 and null set is the smallest subset of every set i.e., ∅ ⊆ A.
Universal Set
If all the sets under considerations or investigation or study are subjects of a fixed set, then the set is called a universal set. It is denoted by U.
Example: In the vowels, consonants studies the universal set consists of all the alphabets in the English Language.
Singleton Set
A set that contains only one element is called a singleton set. It is denoted by {s}.
Example: A = {x : x ∈ N, 7 < x < 9} = {8}
Equal Sets
Two sets A and B are said to be equal and are written as A = B if both have the same elements. This is known as the "Principle of Extension". Therefore, every element which belongs to A is also an element of the set B and every element which belongs to the set B is also an element of the set A.
If A ⊆ B and B ⊆ A ⇒ A = B.
If there is some element in set A that does not belong to set B or vice-versa then A ≠ B, i.e. A is not equal to B.
Example: A = {1, 2, 3}, B = {1, 2, 3} and C = {2, 3, 4}
Here, A and B are equal sets. Also A and C or B and C are unequal sets.
Equivalent Sets
If the cardinalities (number of elements) of two sets are equal, they are called equivalent sets.
Example: A = {1, 2, 3}, B = {1, 2, 3} and C = {2, 3, 4}
Here, A, B and C are equivalent sets.
Disjoint Sets
Two sets A and B are said to be disjoint if no element of A is in B and no element of B is in A.
Example: A = {1, 2 ,3} and B = {4, 5, 6}
Here, A and B are disjoint sets.
Power Sets
The power of any given set A is the set of all subsets of A and is denoted by P(A) or 2A. If A has n elements, then P(A) has 2n elements.
Example: A = {a, b, c}
P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
- A ⊈ P(A) as elements of A (i.e. a, b, c) are not the elements of P(A).
- A ∈ P(A) as A is an element of P(A).
- ∅ ⊆ P(A) as null set is a subset of every set.
- ∅ ∈ P(A) as null set belongs to or is an element of P(A).
- ∅ ⊆ A as null set is the smallest subset of every set.
- ∅ ∉ A as null set is not an element of A.
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