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+10 votes
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The less-than relation, $<,$ on reals is

  1. a partial ordering since it is asymmetric and reflexive
  2. a partial ordering since it is antisymmetric and reflexive
  3. not a partial ordering because it is not asymmetric and not reflexive
  4. not a partial ordering because it is not antisymmetric and reflexive
  5. none of the above
asked in Set Theory & Algebra by Veteran (59.5k points)
edited by | 852 views
0
To be a POSET ...  It hav to be reflexive,Antisyammetric and transitive .... So option E .. correct me if i am wrong ....

3 Answers

+13 votes
Best answer

Relation $<$ is :

  1. not reflexive
  2. Irreflexive
  3. not symmetric
  4. Asymmetric
  5. Anti symmetric


Relation is not POSET because it is irreflexive.
Check AntiSymmetry: $aRb \neq bRa$ unless $a=b.$

A relation may be 'not Asymmetric and not reflexive' but still Antisymmetric.
as $\{(1,1),(1,2)\}$

Not Asymmetric and Irreflexive = Antisymmetric
Option E.

answered by Veteran (55.4k points)
selected by
+2
given relation is not Poset and reason it it is not reflexive..
it has nothing to do with asymmetry.
0
Sir, please let me know why option 'C' cannot be the answer. I'm confused :(
+3 votes

Since  "<" relation neither reflexive nor patialy ordering on set of real number.

But it is anti Symmetric relation.Therefor Option E will be appropriate option for it.

answered by Loyal (8.9k points)
0
Can you explain how it is anti symmeteric?
+1
A binary relation $R$ is anti symmetric if for two elements $x$ and $y$, $xRy$ is true but $yRx$ is not. The binary relation < is anti symmetric because for $x$ and $y$, if $x<y$ is true, then certainly $y<x$ is not.
+1 vote

definitely the less than relation won't be reflexive bcz a<a is always false for every a belongs to real number.

now for antisymmetric  (a<b & b<a) implies a=b , as we know if a<b then b<a can never be true at once therefore         (a<b & b<a) results false. but false can implies anything therefore it is antisymmetric.

for a relation to be POSET: it must be reflexive, antisymmetric and transitive

here it is not reflexive but antisymmetric in nature therefore  only E option is correct.

answered by Active (3.4k points)


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