Find a relation between x which is reflexive, symmetric, but not transitive.

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tiburtius eNotes educator| Certified Educator

Let's define relation `R` on set of real numbers which is both reflexive and symmetric but not transitive.

Let any two real numbers `a`  and `b` be in relation `R` if and only if `|a+b|=|a|+|b|` that is

`forall a,b in RR(aRb)<=>|a+b|=|a|+|b|)`

Reflexivity `aRa?`


Since this is valid for all real number `R` is reflexive.

Symmetry `aRb=>bRa?`

It is easy to see that `R` is symmetrical because of commutativity of addition of real numbers.

Transitivity `aRb ^^ bRc=>aRc`

One counterexample we get for `a=-1`, `b=0` and `c=1`.

`|-1+0|=1=|-1|+|0|` hence `-1R0`

`|0+1|=1=|0|+|1|` hence `0R1`` `

`|-1+1|=|0|=0` and `|-1|+|1|=1+1=2` so

`|-1+1|ne|-1|+|1|` hence -1 and 1 are not in relation `R.`

So we have `-1R0^^0R1` but -1 and 1 are not in relation `R,` thus relation `R` is not transitive over `RR.`