ganeshie8
  • ganeshie8
whats the difference between the function compositions \((f\circ g)\circ h\) and \(f \circ (g\circ h)\)
Mathematics
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SOLVED
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schrodinger
  • schrodinger
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welshfella
  • welshfella
is there a difference?
ganeshie8
  • ganeshie8
It turns out the compositions are associative, but what exactly is the difference between them ?
zzr0ck3r
  • zzr0ck3r
Hint: Function composition is associative

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ikram002p
  • ikram002p
there is no difference only be careful about range and domain
ganeshie8
  • ganeshie8
i mean, in terms of execution order... what difference that parenthesis ordering make ? idk im having a hard time moving forward with my review of abstract algebra because of this..
ikram002p
  • ikram002p
order might affect on that only.
ganeshie8
  • ganeshie8
Hmm okay, domain and codomain are valid in both cases ``` h : X -> Y g : Y -> Z f : Z -> W ```
freckles
  • freckles
hey so you do you mean like what would \[f \circ (g \circ h) \text{ verus what would it look like to find } (f \circ g) \circ h\] \[f=x^2 \\ g=x-1 \\ h= \sqrt{x} \\ f \circ g =(x-1)^2 \\ g \circ h=\sqrt{x}-1 \\ ( f \circ g) \circ h=(\sqrt{x}-1)^2 \\ f \circ (g \circ h)=(\sqrt{x}-1)^2\]?
ganeshie8
  • ganeshie8
I see compositions don't commute in general : \(f\circ g \ne g\circ f\) But I am studying associativity as of now. for example, we knw that addition over real numbers is associative : ``` (a + b) + c = a + (b + c) ``` here we agree that "a+b" is executed first on left hand side. is there a similar thing for compositions ?
ikram002p
  • ikram002p
take this example h : {b,1,2,3}->{1 } g : {1,2,3}->{4,5,a,c} f : {.4,1,2,3}->{a,b,c}
freckles
  • freckles
yes you can find f composed with g first then plug in h or you can find g composed with h then plug that result into f
ganeshie8
  • ganeshie8
how does that work ? say \(f(x) = x\) \(g(x) = 2x\) \(h(x)=3x\)
ganeshie8
  • ganeshie8
\((f\circ g)\circ h(2) = ?\) what does it mean to compose \(f\) and \(g\) first, here ?
freckles
  • freckles
\[(f \circ g)(x)=2x \\ (g \circ h)(x)=2(3x) \\ \text{ now we will replace } x \text{ in first one with } h(x) \\ (f \circ g)(h(x))=2(3x) \\ \text{ now we will replace the } x \text{ in } f(x)=x \text{ with } (g \circ h)(x) \\ \text{ so we have } \\ f((g \circ h)(x))=2(3x)\]
ganeshie8
  • ganeshie8
I think I get it! thanks a lot for breaking it down !
freckles
  • freckles
I kind of like to leave the x off sometimes
freckles
  • freckles
\[(f \circ g) \circ h \\ \text{ where } f=x^2 \\ g=x-1 \\ h=\sqrt{x} \\ f \circ g =(x-1)^2 \\ \text{ then the input is } h \\ (f \circ g) (h)=(\sqrt{x}-1)^2\] it seems less confusing to me for some reason without writing the x
ganeshie8
  • ganeshie8
I see... guess I never played with these much, its kinda fun reviewing/learning these now :) Was going through this proof earlier when I got stuck on compositions : It can be shown that every square matrix represents a composition, therefore the matrix multiplication is also associative!
zzr0ck3r
  • zzr0ck3r
\(f\) is the function \(f(x) = 3x+2\) is the rule \(f(x)\) does not really make sense, but we abuse the hell out of the notation :)
ganeshie8
  • ganeshie8
f : x -> 3x+2 is more proper i think, hope ill be able to appreciate these fine details after finishing abstract algebra xD
ikram002p
  • ikram002p
ur doing abstract now ?
freckles
  • freckles
what?!? you really are doing abstract algebra?
zzr0ck3r
  • zzr0ck3r
\(f:\mathbb{R}\rightarrow \mathbb{R}, f(x) = 3x+2\) is a function
freckles
  • freckles
I thought you already graduated and stuff. You know everything. :p
zzr0ck3r
  • zzr0ck3r
\(f:\underset{x\mapsto 3x+2}{\mathbb{R}\rightarrow \mathbb{R}}\)
zzr0ck3r
  • zzr0ck3r
I really like that last one, but it takes so long in latex
ikram002p
  • ikram002p
hmmm yeah using this notation is better
zzr0ck3r
  • zzr0ck3r
I would not call this abstract algebra. Everything in math is a set. Almost everything is a function.
ganeshie8
  • ganeshie8
lol i suck at math, i am an engineer, what do you think haha! I think abstract algebra is more like pure mathematics , never had to touch this before.
ikram002p
  • ikram002p
nah some times u are perfect in math gane.
ganeshie8
  • ganeshie8
Ahh nice, is it correct to interpret that as : \(f\) is a relation between the sets \(\mathbb R\) and \(\mathbb{R}\) such that \(x\) maps to \(3x+2\)
zzr0ck3r
  • zzr0ck3r
I would say that \(f\) is the name of the function, the name of the rule that gives the relation is \(f(x) = 2x\).
zzr0ck3r
  • zzr0ck3r
so we could say \(f= \{(1,2),(2,4),(4,5)\}\) but \(f(x)= \{(1,2),(2,4),(4,5)\}\) makes no sense.
zzr0ck3r
  • zzr0ck3r
and in this situation we could say \(f(\{1,2,4\})=\{2,4,5\}\).
zzr0ck3r
  • zzr0ck3r
some people would want \(f[\{1,2,4\}]=\{2,4,5\}\)
ikram002p
  • ikram002p
i thought this is also correcto \(f (\{(1,2),(2,4),(4,5)\} )\)
anonymous
  • anonymous
\((f\circ g)(x)=f(g(x))\\((f\circ g)\circ h)(x)=(f\circ g)(h(x))=f(g(h(x)))\\(f\circ(g\circ h))(x)=f((g\circ h)(x))=f(g(h(x)))\)
anonymous
  • anonymous
this is the 'difference' in order, @ganeshie8

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