What is the smallest set that is both 'CLOSED' and for which all of its elements have an 'INVERSE' ?!
God Bless the person who can give me the answer !
2007-11-07 06:35:12 · 1 answers · asked by rOcKsT*R 2 in Science & Mathematics ➔ Mathematics
I'm not sure what you mean by closed. Do you mean topologically closed, or closed under some operation (such as multiplication or addition) and if so, what operation? Similarly, which operation should the set contain inverses for
Assuming you mean closed under some operation, and inverses with respect to the same operation, the answer would ∅, the empty set. This is because the empty set is always closed under any operation, since for a set to not be closed under an n-ary operation, you would have to be able to find n (not necessarily distinct) elements in the set such that the operation on those elements produces an element outside the set. This can't happen, because there are no elements in the empty set. Similarly, every element in the set has an inverse, simply because there are no elements in the set. And of course, the empty set is the smallest set with this property, because it is the smallest set, period.
(Also, this would hold if you meant topologically closed, as ∅ is closed in any topological space).
2007-11-07 06:48:10 · answer #1 · answered by Pascal 7 · 1⤊ 0⤋