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**Sets in Maths **

A set is a grouping of well-defined objects. Only on the basis of simplicity are the objects of a set considered distinct. A family or collection of sets is another name for a set of sets.

Assume we have a family of sets consisting of A1, A2, A3,….. up to An, which is the family A1, A2, A3,….., An and could be denoted as

S = {A_{i} | i belongs to N and 1 ≤ i ≤ n}

A set is denoted by a capital letter and represented by listing all of its elements between curly brackets, such as { }.

## Types of Sets

- Set of singletons

A set only has one element. For instance, A = 3 and B = pencil. Because A and B each contain only one element, they are both singleton sets.

Null Set/Empty Set

An empty set is a set that contains no elements. It is represented by A = { } or A = ϕ.

**Proper set**

If A and B are two sets, then A is a proper subset of B if and only if A ⊆ B but A ≠ B. For instance, if B = {2, 3, 5} , A = {2, 5} is a proper subset of B.

**Power Set**

A set’s power set is the collection of all its subsets. If A is the set, then P(A) is its power set. The number of elements in any power set can be calculated using n[P(A)] = 2^{n}, where n is the number of elements in set A.

- Set that is finite

A set is made up of a finite number of elements.

- Infinite collection

A set is called an infinite set if the number of elements in it is infinite.

**Operations on Sets**

Insets theory, there are three basic operations that can be applied to two sets:

- Combination of two sets
- Two-set intersection
- Difference between two sets

## Relations in Maths

The relation is useful for determining the relationship between a function’s input and output. A non-empty set P to another non-empty set Q relation R is a subset of P X Q.

## Types of Relations

**Empty Relation:**An empty relation (or void relation) is one in which no elements of a set are related to one another.**Universal Relation:**A universal (or full) relation is one in which every element of a set is related to each other.**Identity Relation:**Every element of a set is only related to itself in an identity relation.**Inverse Relation:**When a set contains elements that are inverse pairs of elements from another set, this is referred to as an inverse relation.**Reflexive Relation:**Every element in a reflexive relationship maps to itself.

## Functions in Maths

A function is simply used to represent the dependence of one quantity on another and is easily defined using the mapping concept. In layman’s terms, a function is a relation that produces one output for each input. A function from set P to set Q is a rule that assigns one and only one element of set Q to each element of set P.

A function f from a set P to a set Q, denoted as f: P-> Q, is a mapping of elements of P (domain) to elements of Y (co-domain) in which each element of P is assigned to some chosen element of Q. That is, each element of P must be assigned to at least one element of Q.

**Domain:**

The set of all x values for a function, y = f(x), is known as the domain of the function. It refers to the set of input values that can be used.

Range: The range of y = f(x) is a collection of all f(x) outputs for each real number in the domain. The range is the collection of all y values.

**FAQs**

##### What exactly is a set relation?

A relation between two sets is a collection of ordered pairs that each contains one object from the other set. If object x belongs to the first set and object y belongs to the second set, the objects are said to be related if the ordered pair (x,y) is in the relation.

##### What is the fundamental relationship between sets?

Sets are well-defined collections that are entirely defined by their elements. As a result, two sets are equal if and only if they have the same elements. The fundamental relationship in set theory is that of element hood, also known as membership.