MHB Right inverse clarification needed

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The discussion focuses on the right inverse functions of the function f(x, y) = x + y. Two proposed right inverses, f1(x) = (x-1, 1) and f2(x) = (x-2, 2), correctly satisfy the definition of a right inverse. However, the concern arises regarding the absence of a variable y in these functions, which leads to confusion about defining a more general inverse function like f3(x) = (x-y, y). It is suggested to use a different variable, such as t, to clarify the relationship and avoid confusion with the original function's variables. The overall consensus is that the proposed inverses are valid, but careful notation is essential for clarity.
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f: (R*R)->R
f(x,y)=x+y

if I'm asked to write 2 right inversed fanctions of f.
can I say that:

f1: R-> (R*R)
f1(x)= (x-1, 1)

f2: R-> (R*R)
f1(x)= (x-2, 2)

because: f(f1(x))= f(x-1,1)=x-1+1=x

well this does matches the definition of right inverse function but what bothers me
I guess is that there is no more y on f1.
but it seems more of a problem to define:
f3(x)= (x-y, y)
because then we will receive a hole range of functions. (Whew)

please can someone clarify this point out? thanks so much in advance!(Blush)
 
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dana said:
f: (R*R)->R
f(x,y)=x+y

if I'm asked to write 2 right inversed fanctions of f.
can I say that:

f1: R-> (R*R)
f1(x)= (x-1, 1)

f2: R-> (R*R)
f1(x)= (x-2, 2)

because: f(f1(x))= f(x-1,1)=x-1+1=x

well this does matches the definition of right inverse function but what bothers me
I guess is that there is no more y on f1.
but it seems more of a problem to define:
f3(x)= (x-y, y)
because then we will receive a hole range of functions. (Whew)

please can someone clarify this point out? thanks so much in advance!(Blush)

Hi dana! Welcome to MHB!

That all seems fine to me. ;)

To make the matter less confusing, I would suggest to use a different letter, say t.
So you would have for instance:
$$f_1(t) = (t-1,t)$$
That is be cause this $t$ is in no way connected to the $x$ and $y$ that you have in $f$.

The $y$ that you have in $f_3$ is also not connected to the $y$ in $f$.
It is just an arbitrary constant.
So it would be less confusing to write:
$$f_3(t) = (t - c, c)$$
 
dana said:
f: (R*R)->R
f(x,y)=x+y

if I'm asked to write 2 right inversed fanctions of f.
can I say that:

f1: R-> (R*R)
f1(x)= (x-1, 1)

f2: R-> (R*R)
f1(x)= (x-2, 2)

because: f(f1(x))= f(x-1,1)=x-1+1=x

well this does matches the definition of right inverse function but what bothers me
I guess is that there is no more y on f1.
but it seems more of a problem to define:
f3(x)= (x-y, y)
because then we will receive a hole range of functions. (Whew)

please can someone clarify this point out? thanks so much in advance!(Blush)

You can define an inverse function $\displaystyle x = f^{- 1} (y)$ if it exists a function $\displaystyle y = f(x)$ in which for any value of x there is one and only one value of y. Similarly if You have a two variable function z= f(x,y) You can define two inverse two variable functions $\displaystyle x= f^{-1} (y,z)$ and $\displaystyle y = f^{-1}(x,z)$ if the same conditions are satisfied. In Your case is $\displaystyle z = f(x,y) = x + y$ , so that the two inverse functions are $\displaystyle x = f^{-1} (y,z) = z - y$ and $\displaystyle y = f^{-1} (x,z) = z - x$...

Kind regards$\chi$ $\sigma$
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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