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(Question 2 of Exercise 7.1 Relations in Discrete mathematics and its application by Rosen 7th edition)

The relation below is on the set (1, 2, 3, 4}. Determine whether the relation is reflexive, symmetric, antisymmetric or transitive?

R = {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}

I think the answer must be NO PROPERTY HOLDS FOR THIS RELATION but the answer given in the book is TRANSITIVE. Please explain?

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Yes Transitivity holding here.

Because Transitivity it is telling if a relation holds (a,b) , (b,c) then it must hold (a,c)

But it is not reflexive because (4,4) not in relation

Symmetric property not hold for(2,4),(3,4)

Asymmetric property not hold . Because (2,3),(3,2) holds symmetric property
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R = {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}

From above relation it is clear that it is not Reflexive bcz (1,1) is missing into given relation.

In the case of Transitive We have clear idea that ::

if aRb and bRc then cRa but if aRb and there is no existence of bRa then relation will be Transitive .

Now applying this rule over the pair::
(3,4) and (2,4) so above second bold statement is true so It is transitive.

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