I Circular Reasoning and Impossible Equation

AI Thread Summary
Circular dependencies between variables x and y create a situation where neither can be determined without knowing the other, making it impossible to derive a unique solution from an equation involving both. While equations can be formed, they often result in identities rather than finite solutions, as they may hold true for many ordered pairs. For instance, the equation x = y provides no specific values for either variable despite being logically sound. In contrast, independent equations like x = 2y and y = x - 1 can yield a unique solution. Ultimately, the distinction between unique solutions and infinite solution sets is crucial in understanding these relationships.
e2m2a
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Suppose there is a problem such that in order to know a variable x, you have to know a variable y. But in order to know variable y, you have to know the variable x. Because of this circular dependency, wouldn't it be impossible to write any sensible equation containing x and y?
 
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You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
 
Let'sthink said:
You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
That makes sense. Thanks.
 
What if ##x = 2y## and ##y = x - 1##?
 
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They are two independent equations and will give you the solution for a unique ordered pair (x, y).
 
e2m2a said:
Suppose there is a problem such that in order to know a variable x, you have to know a variable y. But in order to know variable y, you have to know the variable x. Because of this circular dependency, wouldn't it be impossible to write any sensible equation containing x and y?
Take, for example, x=y. From this you can determine neither x nor y. Nevertheless, the equation is very sensible, i.e. contains a lot of useful information. For instance, if x is your position and y is the position of your wallet, and you are a tourist lost in Rio De Janeiro, you will be very happy to know that x=y. :smile:
 
Let'sthink said:
You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
fresh_42 said:
What if ##x = 2y## and ##y = x - 1##?
Let'sthink said:
They are two independent equations and will give you the solution for a unique ordered pair (x, y).
A unique solution is quite different from a solution set that is infinitely large.
 

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