Rewriting empty set

burt

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Technically, does this notation (3,3) mean the same thing as empty set?
 

Romsek

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My first impression would be no, why would it. But maybe you can provide further context.

Oh I guess you mean an open interval... interesting... yes I guess that would be equivalent to the empty set.
 

burt

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pka

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Doesn't that mean the interval from 3 to 3, not including 3?
The statement that \(\displaystyle (a,b)\) is an open interval in the set of real numbers means that \(\displaystyle a~\&~b\) are two real numbers and \(\displaystyle a<b\).
\(\displaystyle (a,b)=\{x\in\mathbb{R}: a<x<b\}\)
 

Romsek

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The statement that \(\displaystyle (a,b)\) is an open interval in the set of real numbers means that \(\displaystyle a~\&~b\) are two real numbers and \(\displaystyle a<b\).
\(\displaystyle (a,b)=\{x\in\mathbb{R}: a<x<b\}\)
and thus \(\displaystyle (a,a) = \{x \in \mathbb{R} : a < x < a\} = \emptyset\)
 

pka

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and thus \(\displaystyle (a,a) = \{x \in \mathbb{R} : a < x < a\} = \emptyset\)
Anyone who can count knows that \(\displaystyle a=a\) so \(\displaystyle (a,a)\) does have two real numbers therefore by definition is not notation for an open interval.
 

burt

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Anyone who can count knows that \(\displaystyle a=a\) so \(\displaystyle (a,a)\) does have two real numbers therefore by definition is not notation for an open interval.
But [a,a] is.
 

Romsek

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