Wikipedia:Reference desk/Archives/Mathematics/2017 July 24

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July 24

Associative property and composite functions

I posted a question a bit back. I have developed some conclusions.

Let $L_{n}=f(T_{n-1})$ and $T_{n}=g(L_{n-1})$, where n is a point in time. Hence,$L_{n-1}=f(T_{n-2})$ and $T_{n-1}=g(L_{n-2})$. Thus, using the substitution property, the composite functions $h_{L}$ and $h_{T}$ can be defined, such that $h_{L}=f\circ g=f(g(L_{n-2}))=L_{n}$ and $h_{T}=g\circ f=g(f(T_{n-2}))=T_{n}$. Finally, due to the associative property, $h_{L}=h_{T}=h$.

I am not certain this line of thinking is correct. I appreciate any insight anyone could provide. Schyler (exquirere bonum ipsum) 17:44, 24 July 2017 (UTC)

I think this doesn't quite work without the assumption that f and g are commutative as well. $g\circ f\circ g\neq f\circ g\circ f$ in general.--Jasper Deng (talk) 18:05, 24 July 2017 (UTC)

Yes, you are confusing associative property and commutative property. Ruslik_Zero 19:55, 24 July 2017 (UTC)

If I am making an argument about human behavior from theory alone, can I assume commutativity? In other words, is it a mathematically sound hypothesis to assume commutativity (I know it is psychologically sound). I am preparing a longitudinal experiment. Schyler (exquirere bonum ipsum) 12:05, 25 July 2017 (UTC)

This is getting to the point where it's very difficult to follow what you're trying to get at unless you're willing to get more specific (didn't I see a guideline or essay or something about this somewhere?). For instance, saying it's "psychologically sound" for two functions to commute with each other just sounds really weird. As for the mathematical validity of assuming it. I mean, sure, but most functions don't actually commute with each other. How about $f(x)=x+2$ and $g(x)=2x.$ Then $f(g(x))=2x+2,$ but $g(f(x))=2x+4.$ --Deacon Vorbis (talk) 15:07, 25 July 2017 (UTC)

I think the OP means "intuitive" when (s)he says "psychologically sound". But as we know, intuition will not fly, and seasoned mathematicians know the necessity of proof before taking a theorem for granted.--Jasper Deng (talk) 17:33, 25 July 2017 (UTC)

Sorry for not being specific enough. I guess it's an intellectual property fear. According to Vygotsky, teaching, $T$, and learning, $L$, are one in the same phenomenon, called obuchenie in Russian and often translated as "teaching/learning" due to difficulty in translation. By "psychologically sound," I mean that theory supports the assumption of commutativity. But I now see that commutativity can absolutely not be assumed in these functions. I have an economic analysis (Correa & Gruver, 1987) that shows commutativity using by a Cournot adjustment, so I was hoping to arrive at a similar conclusion in a different way on theory alone. I am feeling resigned to the need for empirical evidence, though. Schyler (exquirere bonum ipsum) 23:24, 25 July 2017 (UTC)

@Schyler: Well you're not talking about mathematical functions then are you? I would think the psychological equivalent of the noncommutativity is path dependence.--Jasper Deng (talk) 01:24, 26 July 2017 (UTC)