set theory

Question

whether true or false?

If a relation R on set A is irreflexive and transitive then R is antisymmetric.

I am getting false but answer is true..can someone explain with examples

Answer 1

take set A={ 1,2,3}  aRb iff a<b

so define on this relation a set

v={(1,2),( 2,3 ),(1,3) }

so the above set v is ir reflexive and transitive now to check this set is ATS or not

ATS  : (a<b)&(b<a) --------> a= b

CLEARLY LHS of the above implication is false so the above implication become true .  so statement is true ! if u find mistake than let me know .

Comments

your set contain (2,2) which it should not according to definitiin of irreflexive .??

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@kunal

irreflexive: none of the diagonal element should be present

transitive : if (1,2) and (2,3) is present in a relation then (1,3) should be present

let us take an example

R = {(1,2)(2,1)} this is both irreflexive and transitive but not antisymmetric

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Lets say the relation has (2,3)(3,2)(2,2)

the above relation is not reflexive becoz (3,3) is not there ok fine
 but the above relation is not irreflexive becoz above relation contain (2,2) which it should not contain .right !

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actually given answer satisy only one counter examples but it may not cover all the example as such example  given by u above @cse23 so i think its better to leave this question !

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for such question we always search for a counter example, it there is atleast one counter example then we can't ensure such statement as true.

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Hey,

set R = {(1,2)(2,1)} is not transitive.

For it to be transitive you will have to add  {(1,1),(2,2)} to R.

According to me, the given statement is TRUE.

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ya...I think u r ryt..thanks:)
this was the point i was missing
we can have a=c while doing (a,b) and (b,c)
i was taking only distinct values

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Sorry guys. I gave wrong definition of irreflexive relation.