**Definition of Set:** A set is a collection of distinct objects, considered as an object in its own right.

For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a **single set** of size three, written {2,4,6}.

A set is a well defined collection of objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on.

A set is a gathering together into a whole of definite, distinct objects of our perception and of our thought – which are called **elements of the set**.

**Elements of the Set:** Member of the set. It always denoted by small letters.

Example: If “x” is the **element of set** “A” then we can write x is belongs to A.

**Finite Set:** A **finite set** is a set that has a finite number of elements. If the elements of a finite set are listed one after another, the process will eventually “run out” of elements to list.

For example, A = {2, 4, 6, 8, 10}, C = {x: x is an integer, 1 < x < 10}

**Infinite Set:** An **infinite set** is a set which is not finite. It is not possible to explicitly list out all the elements of an infinite set. **Infinite sets** may be countable or uncountable.

Some examples are:

- The set of all integers, {…, -1, 0, 1, 2, …}, is a countable
**infinite set**; and - The set of all real numbers is an uncountable
**infinite set**.

**Unit Set **or** Singleton Set:** A singleton, also known as a unit set, [1] is a set with exactly one element.

For example, the set {0} is a singleton. The term is also used for a 1-tuple (a sequence with one element).

**Empty Set:** The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. It’s denoted by “Ø” or “{}”.

**Null set** was once a common synonym for “**empty set**“, but is now a technical term in measure theory.

**Equal Sets:** Two sets are equal if they both have the same elements.

For example: A = {2, 4, 6} and B = (4, 6, 2) here A = B both are the **equal set**.

**Note:** The order in which the elements of a set are written does not matter.

**Equivalent Sets:** Two sets are equivalent if they both have the same number of elements.

For example: A = {2, 4, 6} and B = {a, b, c} here n (A) = n (G) =3, so set A and set B are equivalent.

**Family of Sets:** A collection of any sets whatsoever is called a **family sets**.

For example: S = {a, b, c, 1, 2}, here S is a **family of sets** over S (in the multi set sense) is given by F = {A1, A2, A3, A4}, where A1 = {a, b, c}, A2 = {1, 2}, A3 = {c, 2}, and A4 = {a, b, 1}.

Here I can say, a collection F of subsets of a given set S is called a **family of subsets** of S. or a **family of sets** over.

**Universal Set:** A **universal set** is the collection of all objects in a particular context or theory including itself. It’s denoted by “U”.

**Sub Sets:** If every **member of set** A is also a **member of set** B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a **super set** of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.

If A is a **subset** of, but not equal to, B, then A is called a proper **subset** of B, written A ⊊ B (A is a proper **subset** of B) or B ⊋ A (B is a proper **super set** of A).

Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).

Example: The set of all men is a **proper** **subset** of the set of all people.

- {1, 3} ⊊ {1, 2, 3, 4}.
- {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The **empty set** is a subset of every set and every set is a **subset** of itself:

- ∅ ⊆ A.
- A ⊆ A.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

- A = B if and only if A ⊆ B and B ⊆ A.

**Power Set:** The **power set** of a set S is the set of all **subsets** of S, including S itself and the **empty set**.

For example, the **power set** of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The **power set** of a set S usually written as P(S).

The **power set** of a **finite set** with n elements has 2n elements. This relationship is one of the reasons for the terminology power set.

For example, the set {1, 2, 3} contains three elements, and the **power set** shown above contains 23 = 8 elements. The **power set** of an infinite (either countable or uncountable) set is always uncountable.

Every partition of a set S is a **subset** of the **power set** of S.