Wikipedia:Reference desk/Archives/Mathematics/2008 May 26

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May 26

Symbols question

What would you use ⋗ / ⋖ for? And what's the difference between ≆ and ≇? Thanks. 70.162.29.88 (talk) 03:49, 26 May 2008 (UTC)

One use of ⋖ and ⋗ is in precedence tables for simple precedence parsers.  --Lambiam 04:21, 26 May 2008 (UTC)

The most common meaning of x ≆ y would be: x ∼ y AND x ≠ y.

The most common meaning of x ≇ y would be: NOT (x ∼ y AND x = y).

I don't remember seeing these in actual use.  --Lambiam 04:35, 26 May 2008 (UTC)

I'd expect x ≇ y to mean 'x is not isomorphic to y' (where the meaning of 'isomorphic' is context-dependent, as always). I've never seen x ≆ y. Algebraist 08:48, 26 May 2008 (UTC)

I'd also go for the idea of not being isomorphic. The first I suppose might mean is not roughly equal to, or maybe is roughly equal to but not exactly. -mattbuck (Talk) 09:53, 26 May 2008 (UTC)

≇ is almost certainly "not isomorphic to". If ≆ means anything, it's "approximately equal but not exactly equal", but I've never actually seen it used. --Tango (talk) 14:01, 26 May 2008 (UTC)

I'm not even sure when you could use such a symbol. If you only know two values approximately, how do you know they aren't exactly equal? Algebraist 14:07, 26 May 2008 (UTC)

Perhaps it would be used in statements such as "If x ≆ 0 then ${\frac {\log(1+x)}{x}}\approx 1$". -- Meni Rosenfeld (talk) 14:51, 26 May 2008 (UTC)

Or, just Pi≆3.14. --Tango (talk) 15:34, 26 May 2008 (UTC)

The LaTeX symbol \cong ($\cong \,\!$) and the HTML entity ≅ (≅) are common notations for is congruent to, as is ∼. I take ≇ to be its negation, which in LaTeX may be denoted \not\cong ($\not \cong \,\!$). Further, presumably ≆ is to ≅ as ⊊ is to $\subseteqq$: the meaning would then be: congruent, but not equal.  --Lambiam 00:52, 27 May 2008 (UTC)

In the Unicode tables, the character ≆ is listed as 'APPROXIMATELY BUT NOT ACTUALLY EQUAL TO' (U+2246), and ≇ as 'NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO' (U+2247). In line with this, ≅ is 'APPROXIMATELY EQUAL TO' (U+2245). In general, whatever the meaning of the latter, it appears reasonable to assume that x ≇ y is equivalent to NOT (x ≅ y) while x ≆ y is equivalent to x ≅ y AND x ≠ y.  --Lambiam 01:06, 27 May 2008 (UTC)

Did you get them backwards above? Surely (x ≇ y) ⇔ ¬(x ≅ y). —Ilmari Karonen (talk) 19:20, 27 May 2008 (UTC)

Yes, thanks. I had them switched but have corrected that now. Somehow these two characters come out so tiny that I can hardly see a difference.  --Lambiam 08:32, 29 May 2008 (UTC)

Why is this notation used?

In the article root (mathematics), the following is used as a definition of what a root is: $x:f(x)=0\,$.

However, while I trust that this notation is correct, I fail to see why the "x :" is needed; it seems redundant to me. --RMFan1 (talk) 19:41, 26 May 2008 (UTC)

I believe I've figured it out: it means "the value of x where...". Correct? --RMFan1 (talk) 19:47, 26 May 2008 (UTC)

It's certainly not "the value of x where...", as there can be many such values or none at all. You're not way off the mark, though - the colon means "such that", that is, "a root of f is an x such that $f(x)=0$". This is usually used in the context of Set-builder notation, though - using it "in the open" like in the article is not common. -- Meni Rosenfeld (talk) 20:09, 26 May 2008 (UTC)

Indeed, it's clearly intended to mean "such that", but "s.t." (or even just "such that") would be the more common notation. I'll change it. --Tango (talk) 20:29, 26 May 2008 (UTC)

I my experience this notation certainly does mean such that and is more commonly used than words but it should be $\{x:f(x)=0\}\,\!$. Another similar notation meaning the same thing you might come across is $\{x|f(x)=0\}\,\!$. Rambo's Revenge (talk) 21:26, 26 May 2008 (UTC)

That's the set of roots, rather than just a definition of a root. --Tango (talk) 19:19, 27 May 2008 (UTC)

Function Graph

If the function $f(x)=\log _{x}(b)$, wherein "x" is variable, and "b" is constant, were to be graphed; what would it look like? Can anyone provide me with a graph of this, please. Thanks, Ζρς ι'β' ^¡hábleme! 21:17, 26 May 2008 (UTC)

Note that $log_{x}(b)={\frac {ln(b)}{ln(x)}}$ or you could use log base 10 instead of natural log if you prefer (in fact any base at all).
You should be able to plot that in some mathmatical software. I can't think of anything online that would graph it, but using Excel or something similar should allow you to get an idea of what it looks like as well. Hope that helps Rambo's Revenge (talk) 21:38, 26 May 2008 (UTC)

There’s quite a few online grapher utilities, such as http://www.walterzorn.com/grapher/grapher_e.htm. Of course you’ll need to know or choose a constant b.GromXXVII (talk) 23:55, 26 May 2008 (UTC)

Of course, log graphs' root is x=1 so, as long as the y-coordinate is not of great importance and the equation isn't in the form $f(x)=\log _{x}(b)+c$, then it's a general log graph. 86.153.37.241 (talk) 23:15, 1 June 2008 (UTC)