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.