Quick convention question

  • #1

Main Question or Discussion Point

Is the closed interval [a,a] considered a legitimate notation for the set {a}? Would (a,a) denote the empty set?
 

Answers and Replies

  • #2
841
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I haven't actually seen anyone use that notation in such a degenerate case, but I don't see any reason why the usage you describe isn't legitimate.
 
  • #3
HallsofIvy
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[a,a] is used- usually in something like [a,b] where you want to consider the possibility that b= a.

If I came across reference to an interval like (a,a), I probably would interpret it as the empty set- although I would wonder why they picked a!
 
  • #4
A={x : f(x)<0 on [a,x]}
Is a in A? It is if f(x)<0 on [a,a]. And since [a,a] has only one number, it suffices to show that f(a)<0.

A={x : f(x)<0 on (a,x)}
Is a in A? It is if f(x)<0 on (a,a). But this is kind of nonsense. There is nothing in (a,a). You make the call.

So anyway, I just wanted to point this out to show why you might write (a,a) rather than {}.
 

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