Introduction to Partial Ordering 

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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. :)

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

use warshall method to solve this by matrices

Had u complete ur undergraduation?

I don’t know but good question, I’m commenting here in case someone answers it.

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.

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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