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Set

Definition

A set is a thing that contains things. They're a pretty fundamental thing in mathematics.

A singleton is a set with exactly one element.

A finite set can be written by listing its elements, e.g. $\{1,2,3\}$ is the set containing the numbers $1,2,3$.

If $x$ is an element of a set $S$, we write $x \in S$.

If $S,T$ are sets, we say $T$ is a subset of $S$, and write $T \subseteq S$ if all elements of $T$ are in $S$; in symbols, $(\forall x \in T: x \in S) \iff T \subseteq S$.

Two sets $S,T$ are equal if they are each a subset of the other one; in symbols, $S = T \iff S \subseteq T$ and $T \subseteq S$

If $T \subseteq S$ but $S \ne T$, we say $T$ is a proper subset of $T$ and write $T \subsetneq S$.

CAVE! Some people use the notation $T \subset S$ to say $T$ is a subset of $S$. Some people use the same notation to say that $T$ is strictly a proper subset of $S$. In these notes, when clarity is needed, I use the unambiguous notation shown above; I sometimes use the $\subset$ notation in cases where it's unambiguous (TODO tidy this up and put an example in an HTML comment) (such as where something's so obviously a proper subset that it feels silly to specify).

These subset and membership symbols can also be written in reverse: $\ni \supsetneq \supset \supseteq$, e.g. $S \ni x$ is equivalent to $x \in S$.

Formal construction

Since I haven't studied set theory in much depth, I can't give a formal construction. You have to be very careful when formalising the notion of a set to avoid problems like Russell's paradox .

Properties

Given sets $S,T$, if

Examples

all these pages adapted with probably insufficient credit from my university's lecture notes