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Question

Is this relation transitive : {(2,7),(3,8),(4,9)]?
(RD sharma says it is transitive)


Solution

Hello,

A binary relation R⊆S×S is transitive if

1.)For alla,b,c ∈ S,[(a,b)∈Rand(b,c)∈R] implies (a,c)∈R .

Inthis case the relation is of the form (x,y) such that x + 5 = y were 1<x<5

Now (a,b) ∈ Rand(b,c) ∈ R will always be FALSE becauseb = 7 in the first case and it cannot be the x (note that x<4) for checking transitivity So the given relation cannot be extended to the form (a,b) ∈ Rand(b,c) ∈ R as b cannot be related to any c and hence the conditio itself can be called psedo true , That is it is transitive The condition to ensure the transitvity of any relation is whether (a,c)∈R.Since within the given range of x no c can be defined we can say it is transitive

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