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Which of the following statements is true?

  1. $(Z, \leq)$ is not totally ordered
  2. The set inclusion relation $\subseteq$ is a partial ordering on the power set of a set S
  3. $(Z, \neq)$ is a poset
  4. The directed graph is not a partial order
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2 Answers

Best answer
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2 votes

A ) (Z,≤) is not totally ordered (false)

The set inclusion relation ⊆ is a partial ordering on the power set of a set S (TRUE)

let S = {a,b}

P(S) = {{a} , {b} ,{a,b} , $\Phi

 

C) (Z,≠) is a poset [False] because Reflexive Relation doesn't satisfy

 

d) The directed graph is not a partial order [FALSE]

 

 

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(1) (Z,≼ ) is a not totally ordered set ------> is false

actually totally ordered set means it should be a poset and each and every two elements are comparable.

http://mathworld.wolfram.com/TotallyOrderedSet.html

 

(2) THe set inclusion relation ⊊ is a poset on the power set of a set S. ---> True

               ( powerset of a set, ⊊ ) ---- satisfies reflexivity, anti-symmetric and transitivity

 

(3) (Z, ≠ ) is a poset ---- false

        it does satisfies symmetric and does not satisfies Anti-Symmetric, counter example is, it contains (2,1) and (1,2)

 

(4) GIVEN DIRECTED GRAPH IS not poset -----> FALSE

         it contains {(a,a),(a,b),(b,b)} --- which is reflexive, Antisymmetric and Transitive ===> it is a poset

 

∴ Option B is true

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