Thread: Language and proofs

I am working on this problem it makes sense to me but there is something inside screaming that it is wrong... here it is.

Consider the equation x^4y+ay+x=0
a) Show that the following statement is false. "For all a,x, in real Numbers, there is a unique y such that x^4y+ay+x=0.

basically i solved for y getting y= -x/(x^4+a). Then explained it as it must be false because x and a are variables spanning R and are conditional to each other so it must be false.

b was just proving that it was true when the equation had to work for x in all real's

Help?? or validate.

Well, you found a counter-example, then it cannot be true.

What if ?

You might be overthinking.... think about this, instead:

Is your expression for y always defined, or are there values of x and a that cause it to diverge?

I'm kind of embarrassed to ask this but does unique mean a single y? i thought it meant that for any value of x and a there was a number y which made the equation true.

Thank you for responding

Originally Posted by j5sawicki

I'm kind of embarrassed to ask this but does unique mean a single y? i thought it meant that for any value of x and a there was a number y which made the equation true.

Don't be embarrassed; you're correct in your original thinking. They seem to be asking whether or not it is true that for any x and a there is a corresponding y such that the equation holds. We have shown that the answer is no, since for certain values of x and a, the equation for y is undefined due to a zero denominator.

There will be many more than just a "single value" for y; the set of y-values should be uncountably infinite.

Hope that makes it somewhat clearer to you.