Naive set theory[242]
As already explained in Definition [1X2], in set theory, the connective "\(\in \)" is added; given two sets \(z,y\) the formula \(x\in y\) reads ”\(x\) belongs to \(y\)” or more simply ”\(x\) is in \(y\)”, and indicates that \(x\) is an element of \(y\).
It is customary to indicate the sets using capitalized letters as variables.
It is usual to write \(x\notin y\) for \(\lnot (x\in y)\), \(x\nsubseteq y\) for \(\lnot (x\subseteq y)\) and so on.
We also define the constant \(\emptyset \), also referred to as \(\{ \} \), which is the empty set, 1 that is uniquely identified by the property
Some fundamental concepts are therefore introduced: union, intersection, symmetric difference, power set, Cartesian product, relations, functions etc.
The power set is defined as in [1Y1].
While in formal theory all the elements of language are sets, in practice we tend to distinguish between the sets, and other objects of Mathematics (numbers, functions, etc etc); for this in the following we will generally use capital letters to indicate the sets, and lowercase letters to indicate other objects.