To prove a proposition is false, do you need to find an explicit example for which the proposition is not true, or can you just assume stuff and show it leads to a contradiction?
For example: Suppose $R_1$ is a total order on $A_1$, $R_2$ is a total order on $A_2$, and $A_1\cap A_2=\emptyset$. Then $R_1\cup R_2$ is a total order on $A_1\cup A_2$.
Now this is a proposition of the form: "for all x, .." right?
So then to show that it is false, you would need to prove that there exists an x such that the statement is false, right?
He is supposed to prove existence of $x$ and $y$, but he starts by saying suppose $x$.., so he basically assumes that such an $x$ and $y$ exist, right? You need to show explicit examples, because that proves existence.
What should I write at the start when proving a proposition is false? If you prove a theorem, you state the premises and the conclusion, and then prove it. But if you are proving a proposition is false, should you say something like: "The proposition X is false". In my version, I state the premises and then a conclusion, but i dont mention anything about a proposition is false, but i would like to.