Question

Just a word on existence vs uniqueness.

Often people are asked to find a solution to a problem.

Example: Find the solution for \(x+7=19\). In most cases you will see the following.

\(x+7=19\)
\(x+7-7=19-7\)
\(x=12\)

So the solution is \(12\).

The problem here is that you did not actually show that the solution to the equation is \(12\). What you have shown is that if it has a solution, the solution must be \(12\). In other words, you have shown uniqueness. To show that \(12\) is a solution, you must actually show it is. i.e. \(19-12=7\) so indeed it is a solution.

Answer 1

So really what you end up turning in is side work, and what we call 'checking our work' is actually answering the question.

Answer 2

I thought that was interesting :)

Answer 3

Isn't x=12 derived logically from the given equation ?
It essentially means than whenever the given equation is is true, the conclusion that x=12 is also true.
Oh the existence part is completely a different thing. It is not necessarily guarenteed by he given eqn hmm

Answer 4

Especially with algebra, there is usually an implication that it is "if and only if" between each line, therefore, at least in @zzr0ck3r 's example, there is also an implication backwards, ie, \((x=12) \Rightarrow (x+7-7=19-7)\) etc Usually, the implication of "if and only if" breaks down when either dealing with inequalities or roots.
tl;dr there is an implicit implication that \(x+7=19 \iff x=12\) so uniqueness and existence is implied... but it is very right to point out there is an implication.

Answer 5

https://akk.li/pics/anne/jpg

Answer 6

Yeah, I guess most students do not think they are doing iff else there would be no need to check the work....

Answer 7

also @ganeshie8 I don't know if a agree that we use iff. The way we teach math we often end up with extraneous solutions and those only fallow logically if we are using strict implies. Right?