Understanding of the word within with regards to translation in math

[Altrepair]

understanding of the word "within" with regards to translation in math

I need a clear understanding of the word "within" with regards to translation in math. Like if I say integers of X is within the interval of 5 and 10. Does this mean: [5,10] (5≤ X ≤10)?? The reason I ask this is because in the English language if I say: "You have within 24 hours to complete the tasks I have given you"; means that I have up to the 24th hour exactly to complete the tasks which means equal to the 24th hour or less.

Here is also a definitions for the word "within".

9.at or to some point not beyond, as in length or distance; not farther than: within a radius of a mile.

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This means I can be equal to a mile in a radius of a mile and still be within a mile or inside the mile radius that is smaller than a mile which is also within a mile. So this means in math that the word "within" always translates to this inequality ≤ right??

 

[haruspex]

Good question!
The meaning in everyday use doesn't help much because nothing is that exact. On the one hand, 'within' does have the sense of 'inside', so would not include the boundaries, but in practice it would be impossible to determine exactly a mile, so whether 'within a mile' includes that is immaterial. It is clearer in the context of a physical boundary: 'within the box'.
In maths, I'm not aware of a standard meaning (nor even of much use of the word). I would always play safe and make it clear whether it is inclusive or exclusive, e.g.: "in the range 5 to 10 inclusive". Or just use the standard notation for a real interval, and inequality symbols for integers.
HTH

 

[NascentOxygen]

Lack of a precise definition means that you should not find yourself having to assign a precise meaning to such a term, IMO. However, I think we'd all be happy to say one throw of a die (dice) must turn up a number within the interval 1..6 so that implies everyday language intends a closed interval for within, at least here.

Not an acclaimed authority, but... http://wiki.answers.com/Q/What_is_an_interval_in_math_term