Introduction to Partial Ordering

Discrete Mathematics: Introduction to Partial Ordering
Topics discussed:
1) Need to study Partial Orderings.
2) The definition of Partial Ordering.
3) Meaning of Poset.
4) Partial Ordering examples.
5) Difference between comparable and incomparable elements.
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Given a set X={2, 3, 4, 5, 6, 7, 8} then divides is a partially ordered relation on X draw the Hasse diagram of POSET where | means divides.

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I dont understand it ,why do you use a,b instead of using some numbers given in the set S in example 1 and 2 aswell. It would be easier for viewers if use explain using those numbers

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[@pranavnyavanandi9710]

Hello @Neso_Academy. I have a doubt. Here's how you defined the meaning of "partial" in Partial-Ordered-Set: The word "partial" in "partial ordering" indicates that not every pair of element in a set is comparable. And comparable means that, the pair of elements are related by the partial ordering relation, that is, "a R b" or "b R a". But then, by this definition every other type of relation, say, an equivalence relation, would also be partial. I am not saying it would be a partially ordered relation but it would be a partial equivalence relation. Why? Because, in an equivalence relation also, not every pair of elements in the set upon which the relation is defined on, are comparable. If this is the case, then partial is not a special property of a poset only but that of every relation since it does not always relate all the elements of a set. Further, every relation does the ordering of elements of a set based on some condition or rule, so what's so special or different about partial ordering? The way I see it, every relation is partial ordering of the elements of a set. Hope I have understood it right. If not, kindly clarify.

[@santerisatama5409]

Partial ordering is defined as 1)Reflexive, 2)Antisymmetric and 3)Transitive. Equivalence relation is symmetric, and thus not partial according to the definition. The term "partial" is used to distinguish from 'totally ordered set', where every pair is comparable, extending the definition with 4) a ≤ b or b ≤ a (strongly connected, formerly called total) to the definition. According to wiki, the informal, intuitive meaning of poset is that "Two elements x and y may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable." Your doubt might originate from this: on a deeper intuitive level, the simplest and most natural definition/derivation or equivalence relation comes from negation of relational operators: if A is neither more nor less than B, then A = B (in a given suitably comparable context). Note that interval from 2 to 4 is both more and less than 3 in the standard ordering of integers. Conditioning by Classical Aristotelean logic (which more or less bans "both-and" and "neither nor") has strong tendency to hide deeply intuitive relations like 'both more and less' and 'neither more nor less' from thinking. Working with intuitive and paraconsistent logics is highly recommended. :)

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Can you give example for better understanding like in this video you gave in 11:35 min.

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(1,1)(2,2)(3,3)(4,4)(6,6)

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What is a poset? Isn't that what we usually have in our sinks where we could turn the water on and off?

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Let A = {1, 2,3, 4} and let R be a relation on A defined by R {(1,1), (1, 2), (2, 4), ,2), (4,3)}. Find the smallest transitive relation R* on A containing . Give explanation of this Question. PLEASE....

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use warshall method to solve this by matrices

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Can any one answer. How many partial order relations are possible on set of n elements?

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[@ahmetkarakartal9563]

I dont understand the result in 11:23 2 divides 1 but the result is not be integer. But in the question I dont see any restriction for this

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please tell i also not understand this relation is not antisymmetric because 1 divide 2 ,2 divide 1 but 1 is not equal to 2 so how is possible to be partial order

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2/1=2 but 1/2 is not equal to integer.

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It's in the definition of "divides". The quotient has to be an Integer. That is, the remainder has to be zero. Otherwise every Integer can of course divide every other Integer.

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5:17 they are not antisymmetric They are Asymmetric.

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