Separate goals from knowns

From CS2800 wiki
(Redirected from Backwards proof)

While writing a proof (as with any technical writing), it is often helpful to remind the reader where you are going. However, it is important to clearly indicate that this is what you are doing.

I often write "we want to show" (abbreviated WTS) to clearly indicate that I haven't already proved what I'm stating. Failing to do so can lead to a backwards proof. For example, while trying to prove [math]A \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math], I shouldn't write something like

  • [math]A \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math]
  • every [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] is in [math]A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math]
  • ...

because the reader will believe that you are claiming what you are trying to prove is true on the very first line (remember, every claim should be obvious based on what came before). However, the following solves this problem:

  • We want to show that [math]A \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math],
  • i.e. that every [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] is in [math]A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math]

Here, each sentence is obviously true: we obviously do want to prove [math]A \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math], and that is obviously the same as saying that every [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] is in [math]A \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} B [/math] (we consider definitions to be "obvious").