Types of Sets
2026-02-28 11:31 Diff
https://brightchamps.com/en-us/math/algebra/types-of-sets

Sets can be divided into various types according to their attributes, based on the elements they contain or the relationship with other sets. Singleton sets, finite and infinite sets, equal and unequal sets, equivalent sets, overlapping and disjoint sets, subsets, supersets, power sets, and universal sets.

Singleton sets: A singleton set is a set that has a single element. It is called a unit set because it contains only one element.

For example, the only number in the set is 6:
\(S = \{6\}\)

Properties of a singleton set:

  1. There is only one element in a singleton set. For example, {6}.
     
  2. Its size, or dimension, is 1. Because it has only one element.
     
  3. A finite set is always a singleton set.

Finite sets: A finite set contains a finite or exactly countable number of elements.

For example, the set \(\{20, 40, 60, 80, 100\}\) contains even numbers. This set has 5 countable elements.

\(S = \{20, 40, 60, 80, 100\}\)

Properties of a finite set:

  1. The set is countable because a finite set has a fixed number of elements.
     
  2. With zero elements, the empty set is also finite.
     
  3. Every singleton set is a finite set, but not all finite sets are singletons.

Infinite sets: An infinite set contains an infinite number of elements.

For example, prime numbers are greater than 1, and they have only two factors: 1 and themselves.

\(D = \{2, 3, 5, 7, 11, 13, 17, 19,...\}\)

The list of prime numbers has no end.

Properties of an infinite set 

  1. It contains many elements that cannot be counted.
     
  2. It is impossible to count every element; it is uncountable.
     
  3. Adding or removing elements does not make a set finite or infinite.

Equal sets: Two sets are equal if they contain the same elements, regardless of what order the elements are in.

For example: \(L = \{blue, violet, orange\}\) and \(D = \{orange, violet, blue\}\) are equal because they contain the same elements.

Properties of equal sets

  1. If every element in one set is also present in the other, then the two sets are equal.
     
  2. In sets, the order of elements can be irrelevant.
     
  3. Two sets are equal when they contain the same elements, even if the order changes.

Unequal sets: Two sets are considered unequal if there is at least one element that differs between them.

Example: Let \(B = \{pineapple, banana, apple\}\) and \(P = \{pineapple, banana, guava\}\). Here, set B and set P are unequal sets because even if one element is different, the sets become unequal.

Properties of unequal sets:

  1. The sets differ, even though one element is not the same.
  2. Sets might or might not contain the same number of elements.

Equivalent sets: When two sets have the same number of elements, even when they are not the same, they are referred to as equivalent sets.

Example: Set \(O = \{2, 5, 6, 7\}\) and \(L = \{w, x, y, z\}\). Since \(n(O) = n(L)\), sets O and L are equivalent in this case.

Properties of equivalent sets:

  1. It has the same number of elements.
     
  2. The number of elements is the same, but the elements themselves differ from each other.
     
  3. Not all equivalent sets are equal, but all equal sets are equivalent.

Overlapping sets: If at least one item from set A appears in set B, then the two sets are said to overlap.

Example: Set \(X = \{2, 4, 6\}\) and \(Y = \{5, 10, 2\}\). In this case, element 2 is present in both sets X and Y. So, they are said to overlap or intersect.

Properties of overlapping sets:

  1. Disjoint sets do not have common elements, and overlapping sets share some elements.
     
  2. If two sets have common elements, they are also known as intersecting sets.
    \(A = \{1, 3, 5, 7, 9\}\)
     

Disjoint sets: If two sets do not have common elements, they are said to be disjoint sets.

Example: Let us assume, set \(U = \{1, 3, 5, 7\}\) and \(S = \{2, 4, 8, 6\}\). The sets U and S are disjoint in this case.

Properties of disjoint sets:

  1. The empty set is the intersection of sets because they don’t have common elements in a disjoint set.

  2. Both finite and infinite disjoint sets are possible.

  3. Even if one or both of the sets are empty, they can still be disjoint.

Subset and superset: If every element in set A is also present in Set B, then set A is a subset of set B.  If set \((A ⊆ B)\), then set B is the superset of set A \((B ⊇ A)\).

Example:
Let \(A = \{10, 50, 30\}\), \(B = \{10, 50, 30, 7,1, 2\}\).

\(A ⊆ B\), since every element in set A is present in set B, \(B ⊇ A\) indicates that set B is set A’s superset.

Properties of subset and superset

  1. Every set is a subset of itself.
     
  2. Supersets can be larger than another set or equal to it.

Universal sets: The collection of all the elements related to a particular topic is known as a universal set.

Example: Let \(P = \{The\ list \ of\ all\ airline \ vehicles\}\) that contains helicopters, jets, and rockets. In this case, if P is the set of all airline vehicles, then jets, helicopters, and rockets are subsets of P because they are the types of airline vehicles.

Properties of universal sets:

  1. A universal set can be finite or infinite, depending on the context.
     
  2. All sets are subsets of the universal set.
     
  3. To find a set’s complement, subtract its elements from the universal set.
     

Power sets: A power set is the collection of all possible subsets within a set.

Example: All possible subsets of A are included in the power set of \(A = \{m, n\}\), which is \(P(A) = \{∅, \{m\}, \{n\}, \{m, n\}\}\).

Properties of power sets:

  1. The power set of set A contains all possible subsets of A.
     
  2. A set with n elements will have 2n subsets in its power set.
     
  3. The power set always has more elements than the original set if the original set is not empty.