I found the answer the question i post.
Answer is " Does not exit"
since supremum and infimum are exits only for real numer system,
but in my question that set contains only set of all rational numbers between ?2 and ?3.
Therefore supremum and infimum does not exits for that set.
You have to consider the set as a subset of the real numbers. It is quite possible that the sup and inf are not elements of the set itself, and any subset of the reals that is bounded above (resp. below) has a supremum (resp. infimum).
Sure the set has no supremum or infimum in the rational number system, but if that was meant to be answered within the rationals, it really shouldn't have been defined as the set of rationals between "sqrt2 and sqrt3", but rather the set of "positive rationals x with x^2 between 2 and 3", since several logical and philosophical objections arise automatically from the former verbatim.
Considering this I personally think the set was meant to be viewed as a subset of real numbers, and not in the rational system. In this case the infimum and supremum sure dn exist and finiding them is not much hard.
However, I suggest you should refer to the teacher/ professor/ textbook that had proposed the question for an explicit refrence to the system on which the exterma are to be found.
Maybe, and sure IS odd in general, but some special instances are quite commonplace, like the classical counter-example of the set of rational numbers r with r^2<2 used by many textbooks to prove that the ordered field of rationals is not complete.