Table of Contents

A set is indeed 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 yet another name for a set of sets. Assume we have a family of sets consisting of A_{1}, A_{2}, A_{3},….. up to A_{n}, which is the family {A_{1}, A_{2}, A_{3},….., A_{n} } and could be denoted as:

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

**Types of Sets**

**Singleton set:** The set which only has one element.

**Empty Set/Null Set:** An empty set is a collection that contains no elements.

**Proper set:** If indeed A and B are two sets, then A is said to be a proper subset of B if A ⊆ B but A ≠ B.

**Power Set:** The power set of a set is really the collection of all its subsets. If A is indeed the set, then P(A) is its power set.

The number of items in any power set can be calculated using n[P(A)] = 2^{n}, where n is the number of elements in set A.

**Relations**

The relation would be 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) would be one in which no elements of a set are related to one another.

**Universal Relation: **A universal (or full) relation would be one in which every element of a set is related to each other.

**Identity Relation:** In quite an identity relation, every element of a set is only related to itself.

**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:** Each element in a reflexive relationship maps to itself.

**Symmetric Relation:** When a=b is true, then b=a is also true in an asymmetric relationship.

Transitive Relation: For this relation, when (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R.

**Equivalence Relation:** An equivalence relation is something that is reflexive, symmetric, and transitive all at the same time.

**Functions**

A function has been simply used to represent the dependence of one quantity on another and is easily defined using the mapping concept. In basic terms, a function is a relation that produces one output for each input.

One function from set P to set Q is indeed a rule that assigns one and only one element of set Q to each element of set P.

**Domain, Co-domain and Range of a Function**

A function is a mathematical relation between two sets, usually denoted by an equation. The function assigns a unique output to every input. The input and output can be real numbers, but they can also be complex numbers.

The domain of a function is the set of all inputs for which the function produces a result. The co-domain is the set of all outputs for which the function produces a result. The range is the set of all outputs that the function can produce.

For example, the domain of the function f(x) = x2 is all real numbers. The co-domain is all real numbers. The range is all real numbers except for negative numbers.

The domain of the function g(x) = x3 is all real numbers. The co-domain is all real numbers. The range is all real numbers except for negative numbers and zero.

The domain of the function h(x) = x is all real numbers. The co-domain is all real numbers. The range is all real numbers except for zero.

**FAQs**

##### What is set relation?

A relation between two sets would be a collection of ordered pairs that each contains one object from the other set.

##### How do you tell if a set of relations is a function?

When a set of ordered pairs passes the vertical line test, it is a function. Since there is only one corresponding value for any given value, the ordered pair relation IS a function.