In these illustrative examples, some have mistakenly been using the tilde ("~") to express negation, and a few ignorant brutes have been using the exclamation point ("!") as well. The latter I will not deign to address, as it is clearly an abomination, but the tilde requires some discussion.
As anyone familiar with mathematical journals or better textbooks can tell you, the negation symbol ("¬") is quite distict from the tilde. The tilde, a mere punctuation mark, meaningless on its own, was originally introduced into some inferior mathematical literature solely as a consequence of inadequate typesetting capabilities.
The days of such technical problems are long gone; if you wish to use the negation symbol today, all you need to do is locate the appropriate key or switch on your writing mechanism, and activate it. I will illustrate with an example, but first I would like to remind everyone that mathematical examples should be clearly set apart from prose using appropriate spacing. The occasional exception will of course crop up, to the amusement of all, but please try to be considerate, and keep the sensibilities of your readers in mind. Now, we will illustrate with that old chestnut, the contrapositive:
if a then b <-> if ¬b then ¬a
I am sure this reminder is not necessary for most of you; for the rest, please do not take this as an insult; after all, anyone might lose sight of details like this from time to time.