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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?
The empty set supremum proof is a mathematical proof that shows that the empty set, denoted by ∅, has a supremum (least upper bound) of -∞. This proof is important in understanding the properties of sets and their elements.
The empty set supremum proof is significant because it helps establish the properties of the empty set, which is a fundamental concept in mathematics. It also allows for the application of set theory in various fields, such as computer science and statistics.
The empty set supremum proof is derived using the definition of supremum, which states that a supremum of a set S is the smallest upper bound of S. Since the empty set has no elements, it is vacuously true that all real numbers are upper bounds of the empty set. Therefore, the supremum of the empty set must be the smallest of these upper bounds, which is -∞.
The implications of the empty set supremum proof are vast and far-reaching. It helps establish the foundations of set theory and provides a basis for understanding more complex mathematical concepts. It also has practical applications in fields such as computer science, where the empty set is used to represent an absence of data.
One common misconception about the empty set supremum proof is that it implies that the empty set is equal to -∞. However, this is not the case. The proof simply shows that -∞ is the supremum of the empty set, but the two are not equivalent. Another misconception is that the empty set has no properties, but the empty set supremum proof shows that it does have the property of having -∞ as its supremum.