In this article, we will discuss Types of Sets in Discrete Structure or Discrete Mathematics. Also, we will cover the examples. Let’s discuss one by one.

**Partially Ordered Set or POSET :**

Partial Ordered Set (POSET) consists of sets with three binary relations as follows.**Reflexive Relation –**One in which every element maps to itself.**Anti-Symmetric Relation –**If (a, b) ∈ R and (b, a) ∈ R, then a=b.**Transitive Relation –**If (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R).

**Examples –**

Let A be a set : A = {1, 2, 3}.Question-1 : R1 = { } . Is R1 a POSET?

**Answer –**

R1 is not a POSET because R1 is not reflexive. If any of the three relations is not available then it is not a POSET.Question-2 : R2 = {(1, 1), (2, 2), (3, 3)} . Is R2 a POSET?

**Answer –**

This is a POSET because R2 is reflexive, transitive as well as anti-symmetric.

Question-3 : R3 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} . Is R3 a POSET?

**Answer –**

This is not a POSET because R3 is reflexive but not anti-symmetric. It is actually symmetric. So it cannot be a POSET.**Linearly Ordered Set :**

It is also known as**Chain or Totally Ordered Set.**It is basically a POSET in which given any pair (x, y) satisfies either x ≤ y or y ≤ x. Or we can say that if any one of the statements “x<y, y<x, x=y” is correct (This is also called as*Law of Trichotomy*) then it is a Linearly Ordered Set.**Example –**

A set of real numbers with a natural ordering ([R, ≤]) is a Linearly Ordered Set because firstly it is a POSET, and if we take any two real numbers e.g. (r1, r2) ∈ R then at least any one of the following three statements is always true: r1<r2 or r2<r1 or r1=r2. So, it is a Totally or Linearly Ordered Set.**Isomorphic Ordered Sets :**

Let (A, ≤) and (B, ≤) be two partially ordered sets then they are said to be isomorphic if their “structures” are entirely similar.**Example –**

Let two POSETS, A = P({0, 1}) ordered by ≤ and B = {1, 2, 3, 6} ordered by division relation are Isomorphic Ordered Sets.**Explanation :**

**Hasse Diagram of POSET A –**A = { Φ, {0}, {1}, {0, 1} } with subset relation.

**Hasse Diagram of POSET B –**

If we try to define a map.

f( Φ ) = 1, f( {0} ) = 2, f( {1} ) = 3 and f( {0, 1} ) = 6. So both the sets are isomorphic. Hence, they are Isomorphic Ordered Set.

**Well Ordered Set :**

A partially ordered set is called a Well Ordered set if every non-empty subset has a least element.**Example –**

A set of natural number and less than operation ([N, ≤]) then it is a Well ordered Set because firstly it is a POSET and if we take any two natural numbers e.g. n1 and n2 where n1≤n2. Here, n1 is the least element. So, it is a Well Ordered Set.**Note –**- Any Well ordered set is totally ordered.
- Every subset of a Well ordered set is Well-ordered with the same ordering.

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