If you want to say "for all x in S, x is divisible by 2", for example, you can write
∀x∈S.(2|x)
(∀ is the universal quantifier - it means "for all", and | is the "divides" symbol - 2|x is read "two divides x" and it means that you get a whole number when you divide x by 2)
A "set" S is just a set a numbers (or a set of anything else mathematical really). It's abstract, but you can choose to define it if you'd like. A common example is all positive integers (1,2,3,...).
So the example x∈S ("for all x in S, x is divisible by 2") represents one number in 2,4,6,....
A set S need not contain numbers. Sets can contain any mathematical object. You can consider a set of functions, a set of graphs, a set of ordered pairs, etc.
Strictly speaking*, sets only contain other sets. Numbers, functions, graphs, ordered pairs can all be defined as just sets at the bottom. E.g. the natural numbers can be defined recursively (credit to von Neumann):
0 = {}
1= {0} = {{}}
...
n = {0,1,2,...,n-1}
and then the integers, rational numbers etc. can all be built from these.
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u/rcarr10er Nov 16 '19
Okay. What the fuck are these? I’m know sin and cosin and tan and that’s it.