Terms involving sets

it's time to begin applying the language and logic of to proff writing. A good place to do this is with sets. in this section, we addres two things. first, we return to the set terminology and we use it to practice some of the concepts we've learned so far. second, we get our feet wet by beginning to write proofs. Right off the bat, we'll see there useful techniques for writing proofs: direct proofs; proofs by contrapositive; and proofs by contradiction.
Terms Involving sets
define the word definition. the self-referencing nature of this request might make is seem like a hard thing to do, but surely we would agree that a basic feature of a definition is that it gives us a way of subtituting a single word for a whole phrase. the single word being defined is created for the express purpose of being equivalent to that phrase. you could say:
Jessica went to kenya and contracted one of a group of diseases, unsually intermittent or remittent, characterized by attacks of chills, fever, and sweating, caused by a parasitic protozoan, which occupies and destroys red blood cells.
Or you could say simply that Jessica went to Kenya and contracted malaria. Malaria is defined to be that disease. you can get wothout the word malaria if you want, but then you have to use a whole lot of words every time you want to address this disease (which we hope is not very often), and you won't get practice in logic. for, you see, a definition is an axiomatic statement of logical equivalence. in this case, we're declaring that the statement
x is malaria
x is disease characterized by ....
ia a tautology. so, if x is malaria, then x is a disease with those scary symtoms, and if x is a disease with those scary symptoms, then x is malaria.