- #1
Mr Davis 97
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I'm trying to really get a grasp on proofs of uniqueness.
Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##.
First method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: If ##ax+b=0##, then ##ax = -b## and so ##x = -b/a##.
Second method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: Suppose that ##y## and ##z## are two different solutions to the equation. Then ##ay+b=0## and ##az+b=0##, so then ##ay+b=az+b##, which implies that ##y=z##.
What is the difference between these two methods? Are they both valid? Which is preferable?
Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##.
First method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: If ##ax+b=0##, then ##ax = -b## and so ##x = -b/a##.
Second method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: Suppose that ##y## and ##z## are two different solutions to the equation. Then ##ay+b=0## and ##az+b=0##, so then ##ay+b=az+b##, which implies that ##y=z##.
What is the difference between these two methods? Are they both valid? Which is preferable?