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evagelos
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Does the empty set have a supremum ( least upper bound)? if yes, can anybody give me a proof please? if no, again a proof please?
Focus said:By convention it is [tex] -\infty [/tex]. No proof :)
mathman said:Since the set is empty, it really has no bounds. Therefore one can prescribe bounds by convention.
peos69 said:IS that new mathematics??
peos69 said:Sure when we cannot prove something we use the convention stuff
peos69 said:The empty set is bounded
peos69 said:The question is asking for the least upper bound not just for bounds
Focus said:If a subset of R is bounded then a supremum exists by the completeness of R. There is no bound and a supremum does not exist for the empty set.
LukeD said:Anyway, the completeness axiom only says that non-empty subsets of R with upper bounds have least upper bounds. It doesn't say anything about the empty set, and it's easy to prove that it does not have a least upper bound.
peos69 said:Besides that is a semantical proof based simply on the F----->T truthfulness
In a real proof which is syntactical the words true false are not used.
hence the proof that the empty set is bounded from above
is......pending
LukeD said:Focus: The definition of a bound (at least the one I've been taught) is that M, a real number, bounds S, a subset of the real numbers, if for all x in S, |x| <= M (and you can also define upper bound and lower bound. Clearly, a set that is bounded has both upper and lower bounds)
Since the empty set has no elements, the statement "for all x in the empty set, |x| <= M" is vacuously true no matter what M is. Therefore, every real number is a bound of the empty set.
But that's the point -- the statement "for all x in the empty set <blah blah blah>" is vacuously true no matter what, since there is no x in the empty set.Focus said:There is no x an element of the empty set. Thats a bit of a contradictory statement to make. I am not worried about the M part, its the bit that says for all x in empty set.
morphism said:But that's the point -- the statement "for all x in the empty set <blah blah blah>" is vacuously true no matter what, since there is no x in the empty set.
Focus said:Hmm sorry my bad. Might be more useful to define it like for all x, x in empty set implies x is less or equal than M.
LukeD said:.
Proof:
Assume for sake of contradiction that the empty set has a least upper bound, we'll call it u. u-1 also bounds the empty set (since every real number bounds the empty set), so it is an upper bound. However, u-1 < u, which is the least upper bound. This is a contradiction, and therefore, the empty set has no least upper bound.
evagelos said:since u is the least upper bound we have u<u-1 and not u-1<u so where is the contradiction
LukeD said:Skimmed the Wikipedia article a bit. It seems pretty good.
http://en.wikipedia.org/wiki/Vacuous_truth
LukeD said:.
Proof:
Assume for sake of contradiction that the empty set has a least upper bound, we'll call it u. u-1 also bounds the empty set (since every real number bounds the empty set), so it is an upper bound. However, u-1 < u, which is the least upper bound. This is a contradiction, and therefore, the empty set has no least upper bound.
evagelos said:I am sorry to say nobody yet has given me the definition of the 'Vacuously true' expression
evagelos said:I am sorry to say nobody yet has given me the definition of the 'Vacuously true' expression
LukeD said:Since the empty set has no elements, the statement "for all x in the empty set, |x| <= M" is vacuously true no matter what M is. Therefore, every real number is a bound of the empty set.
Yes, but wording it "If x is in the empty set then |x|<= M" gives a valid, vacuously true statement.Focus said:There is no x an element of the empty set. Thats a bit of a contradictory statement to make. I am not worried about the M part, its the bit that says for all x in empty set.
evagelos said:Give me a definition of the 'Vacuously true' expression please
In logic the statement A=> B or "If A then B"is true if A is false no matter whether B is true or false. That is called "vacuously true".evagelos said:I am sorry to say nobody yet has given me the definition of the 'Vacuously true' expression
evagelos said:Thanks:
can this 'Vacuously true' expression' be considered as an axiom, a theorem, or what?