Timeline for Examples of common false beliefs in mathematics
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 26, 2014 at 4:12 | comment | added | Bruce Blackadar | Here is a related but slightly less obvious situation. The ordered pair $(a,b)$ is generally defined in set theory to be $\{\{a\},\{a,b\}\}$. This is generally thought of as a set with two elements. But what if $a=b$? | |
| Dec 15, 2013 at 5:16 | comment | added | Toby Bartels | In a context where one is discussing real analysis, $e$ and $\pi$ are generally taken to be the famous constants. But this is hardly universal; in other contexts, they may have very different meanings. | |
| Dec 14, 2013 at 22:26 | comment | added | LSpice | @TobyBartels, I think that this statement can be believed only if one is willing to speak of the 2-variable polynomial $e\pi$. (Perhaps you regard $e$ and $\pi$ as part of a universal context. :-) ) | |
| Apr 4, 2011 at 9:41 | comment | added | Toby Bartels | Single-letter symbols are usually assumed to be variables, if the context doesn't determine otherwise, even in the absence of quantifiers. (You can put in an implicit universal quantifier to close up all sentences.) | |
| Dec 1, 2010 at 19:35 | comment | added | roy smith | E.g. if you said something like "for all a,b, (in some given universe) the set {a,b} has two elements", then I would agree. | |
| Dec 1, 2010 at 19:34 | comment | added | roy smith | I am confused here, as there is no suggestion that a,b are variables, since there are no quantifiers. It seems to me this is true since the first two letters of our alphabet are indeed distinct. | |
| Jun 17, 2010 at 23:34 | comment | added | Daniel Asimov | There are many situations where one needs to speak of a set of two numbers that may or may not be equal. E.g.: "Let x<sub>1</sub>, x<sub>2</sub> ∈ ℝ. Then among all the open intervals containing the set {x<sub>1</sub>, x<sub>2</sub>}, none of them is contained in all the others." If one is addressing mathematicians, there is no need to specify that x<sub>1</sub> might be equal to x<sub>2</sub>. | |
| May 6, 2010 at 11:16 | comment | added | David E Speyer | I'm not sure there is a false belief here, as much as awkward writing. Depending on context, I might very well write "The set $\{a,b \}$ (where $a$ and $b$ might be equal)..." if this issue mattered. | |
| May 6, 2010 at 10:58 | history | answered | Tom Pinkeith | CC BY-SA 2.5 |