# Checking wether a 2 variable 2 valued function is bijective

• Jan 22nd 2006, 07:00 PM
SkanderH
Checking wether a 2 variable 2 valued function is bijective
Hi everybody

Could anyone please tell me how to check or prove if a function from [0,1]^2 to [0,1]^2 is a bijection. It easy from [0,1] to [0,1], but not so obvious in the 2 dimensional case. The type of function I am thinking of is continuous but not necessarly differentiable (e.g I don't want to have to use partial derivatives)

thanks
• Jan 22nd 2006, 07:05 PM
ThePerfectHacker
What is ^2? What type of function is that?

Use the mathematical meaning of bijective.
I do not want to get in to the whole definition of a function from a set-theoretic viewpoint, but the basic thing you need to show 2 things.

Step 1:One to One Show that $\displaystyle f(x_1,y_1)=f(x_2,y_2)$ if and only if $\displaystyle (x_1,y_1)=(x_2,y_2)$

Step 2:Surjective Show that for any $\displaystyle r\in R$ (any real number) the equation $\displaystyle f(x,y)=r$ has a solution.

Sometimes showing part 2 is more difficult, because you prove existence for that part. Sometime what you can do is prove that the function is countinous, then by the intermediate value theorem it must have a solution.

Of course, if you need to show that a function is bijective in a disk then do the same steps but only limited to that disk.
• Jan 22nd 2006, 09:25 PM
SkanderH
by [0,1]^2, I meant [0,1] to the power of 2, the unit square.
• Jan 23rd 2006, 11:02 AM
ThePerfectHacker
Any specific problem?
• Jan 23rd 2006, 04:54 PM
SkanderH
Yes

I am trying to find a Fuzzy logic version of reversible boolean logic gates like the Toffoli gate or the Fredkin gate.

I am first trying the most basic reversibel gate, the C-not gate, which in boolean maps (a,b) to (a,a xor b), but all T-norm and S-norms I used produce irreversible gates.
The fuzzy version should map from the unit square to the unit square again. I don't if the problem is with my choice of T-noms, or is the problem inherent in the characteristic of C-not.

http://en.wikipedia.org/wiki/Reversible_computing
http://en.wikipedia.org/wiki/Toffoli_gate
http://en.wikipedia.org/wiki/Fredkin_Gate
http://en.wikipedia.org/wiki/T-norm
• Jan 23rd 2006, 06:00 PM
ThePerfectHacker
Quote:

Originally Posted by SkanderH
Yes

I am trying to find a Fuzzy logic version of reversible boolean logic gates like the Toffoli gate or the Fredkin gate.

I am first trying the most basic reversibel gate, the C-not gate, which in boolean maps (a,b) to (a,a xor b), but all T-norm and S-norms I used produce irreversible gates.
The fuzzy version should map from the unit square to the unit square again. I don't if the problem is with my choice of T-noms, or is the problem inherent in the characteristic of C-not.