Difference between revisions of "Infinite"
ComplexZeta (talk | contribs) |
(→Operations with Infinity) |
||
| (7 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
| − | A set <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. | + | A [[set]] <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. |
| + | |||
| + | In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that you haven't examined yet. | ||
| + | |||
| + | ===Equivalent formulations=== | ||
| + | * A set is infinite if it can be put into [[bijection]] with one of its proper [[subset]]s. | ||
| + | * A set is infinite if it is not empty and cannot be put into bijection with any set of the form <math>\{1, 2, \ldots, n\}</math> for a [[positive integer]] <math>n</math>. | ||
| + | |||
| + | ==Applications to Infinity with Sums== | ||
| + | |||
| + | A sum works the same way. Certain sums equate to infinity, such as | ||
| + | |||
| + | <math>\sum_{i = 3}^{\infty}{(2i - 1)}</math> | ||
{{stub}} | {{stub}} | ||
| + | |||
| + | |||
| + | =="Operations" with Infinity== | ||
| + | |||
| + | Some '''bad''' rules involving operations with infinity are as follows: | ||
| + | |||
| + | * <math>1/{\infty} = 0</math> | ||
| + | |||
| + | * <math>{\infty} + x = {\pm}{\infty}</math> | ||
| + | |||
| + | * <math>{\infty}\cdot{x} = {\infty}</math> | ||
| + | |||
| + | None of these are true because <math>\infty</math> is not a real number which you can write equations involving. | ||
Latest revision as of 22:19, 16 August 2013
A set 
is said to be infinite if there is a surjection 
. If this is not the case, is said to be finite.
In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that you haven't examined yet.
Equivalent formulations
- A set is infinite if it can be put into bijection with one of its proper subsets.
- A set is infinite if it is not empty and cannot be put into bijection with any set of the form
for a positive integer 
.
for a 
.Applications to Infinity with Sums
A sum works the same way. Certain sums equate to infinity, such as
This article is a stub. Help us out by expanding it.
"Operations" with Infinity
Some bad rules involving operations with infinity are as follows:
None of these are true because 
is not a real number which you can write equations involving.
