What I liked and didn’t like about the set theory so far

The idea of two sets being equivalent is both helpful and confusing. I have to remind myself that equivalent sets are not necessarily equal sets. And this is the major source of confusion. Sets that have the same number of members are equivalent. This means that the determinant is the cardinality of sets that we are comparing, not their content. A funny example may be that the set of meals one might have in a day {breakfast, lunch, dinner} is equivalent but not equal to the set of arms on a clock {hour, minute, second}. I want to think that logical thinking requires that math (the language of science) provide a comparison system beyond equality. And the concept is probably not very complicated if I look at it from correct perspective. If I have a set of jet fighters and a set of fighter pilots comparing the individual members of each set with each other may be useless. But knowing that the two sets are equivalent allows me to say that I have a pilot for every jet, and that is a useful statement.

Uncommon symbols and their meaning

Last week’s lecture started discussion of logic and in designated “~” as the symbol for negation. I knew that in various programming languages “!” is used to refer to the opposite of a value, as in: if ! (foo.class = bar) then quz. So my mind wondered about different math symbols and I remembered an old question I had been meaning to look up. Namely, the difference between two stroke equal signs and three stork equal sign. A few google searches provided the answer. The difference between the two symbols only becomes evident when a variable is present in the statements involving the equal sign. The three stroke equal sign is used when the statement is valid regardless of what we replace the variable with. And the two stroke equal sign is designated for situations where only a limited set of values can replace the variable.
(x+2)^2≡x^2+4x+4 is valid for any replacement of x whereas (x+2)^2=16 is only valid if x is 2 or -6. So, the three stroke equal sign designates an identity, more on that later.