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+8 votes
$S=\{(1,2), (2,1)\}$ is binary relation on set $A = \{1,2,3\}$. Is it irreflexive? Add the minimum number of ordered pairs to S to make it an equivalence relation. Give the modified $S$.

Let $S=\{a,b\}$ and let $\square(S)$ be the powerset of $S$. Consider the binary relation '$\subseteq$ (set inclusion)' on $\square(S)$. Draw the Hasse diagram corresponding to the lattice ($\square(S), \subseteq$)
asked in Set Theory & Algebra by Veteran (68.8k points)
edited by | 537 views

2 Answers

+12 votes
Best answer

S={(1,2), (2,1)} -> This relation is Irreflexive, Symmetric, Not Transitive, Not Reflexive, Not Asymmetric, Not antisymmetric.

Equivalence Relation -> Symmetric, Transitive, Reflexive.

It is not transitive & Reflexive.

So Reflexive closure of S = { (1,1),(2,2),(3,3),(1,2),(2,1) }

After taking transitive closure relation does not change.

Answer S = { (1,1),(2,2),(3,3),(1,2),(2,1) }

S = {a,b}

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

Related Hasse Diagram


answered by Veteran (48.6k points)
edited by
+4 votes
S={(1,2), (2,1)} is irreflexive as well as symmetric but not transitive..
to make equivalence relation we need to add Reflexive, symmetric, transitive closure to it.
reflexive closure = {(1,1,) (2,2) (3,3)}
Symmetric Closure = { }
Transitive Closure = {1,1 }
equivalence relation = {(1,1,) (2,2) (3,3) (1,2) (2,1)}
answered by Veteran (49.2k points)
edited by
Hi @laser , is the original relation transitive ? ( i am not talking about modified relation ).

In the original relation , { (1,2), (2,1) } : it is (a) irreflexive , (b) symmetric , but is it transitive ?

I know null set is also transitive . But , in the given original set had there been an element (1,1) , it would have been transitive. Am I wrong ?
corrected !!!

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