Common Notations:

\(\mathbb{N}\): The set of natural numbers \(1,\ 2,\ 3,\ \dots\) and so on. It is a matter of style whether or not to include \(0\) in \(\mathbb{N}\), we shall do so unless explicitly stated otherwise.

\(\mathbb{Z}\): The set of positive and negative whole numbers, \(\dots, -2,\ -1,\ 0,\ 1,\ 2,\ \dots\)

\(\mathbb{Q}\): The rational numbers, e.g. the numbers which can be represented as a fraction of two integers where the denominator is nonzero.

\(\mathbb{R}\): All real numbers.

Informal Notions:

Set: A collection of things.

Empty Set: The unique set with no elements, denoted by \(\emptyset\).

Set Builder Notation: If \(P(x)\) is some statement about elements of a set \(Y\), then \(\{x\in Y:P(x)\}\) is the set containing all objects \(x\in Y\) that satisfy \(P(x)\). As an example, if \(P(x)\) says “\(x\) is even”, then \(\{x\in\mathbb{N}:P(x)\}\) is the set of even natural numbers.

Subset: If \(X\) and \(Y\) are sets, then \(Y\) is a subset of \(X\), written \(Y\subseteq X\), if everything in \(Y\) is in \(X\).

Elements: If \(X\) is a set and \(a\) is something inside the collection represented by \(X\), we say \(a\) is an element of \(X\) and write \(a\in X\).

Power Set: If \(X\) is a set, then the power set of \(X\), written \(\mathcal{P}(X)\), is the set whose elements are exactly the subsets of \(X\).

Union: If \(A\) and \(B\) are sets, then \(A\cup B\) denotes their union, the set of all elements in at least one of \(A\) or \(B\). If \(T\) is a set of sets, then \(\bigcup_{A\in T} A\) denotes the union of all the elements of \(T\), i.e. \(a\in \bigcup_{A\in T} A\) if and only if \(a\in A\) for some \(A\in T\).

Intersection: If \(A\) and \(B\) are sets, then \(A\cap B\) denotes their intersection, the set of all elements in both\(A\) and \(B\). If \(T\) is a set of sets, then \(\bigcap_{A\in T} A\) denotes the intersection of all the elements of \(T\), i.e. \(a\in \bigcap_{A\in T} A\) if and only if \(a\in A\) for all \(A\in T\).

Set Equality: Two sets are considered equal if they have the same elements. That is, if every element of \(A\) is in \(B\) and every element of \(B\) is in \(A\), then \(A=B\). Equal sets are indistinguishable, so if \(A=B\) and \(A\in C\), then \(B\in C\).

Function: A function \(f:X\to Y\) is a set of pairs \((x,y)\) such that \(x\in X\), \(y\in Y\), and for every \(x\in X\) there is a unique \(y\in Y\) such that \((x,y)\in f\). If \((x,y)\in f\), then we say \(f(x)=y\). Note that the \(y\) only needs to be unique for a fixed \(x\), we can choose the same \(y\) for multiple \(x\)’s but only one \(y\) per \(x\).

Inverse Image: If \(f:X\to Y\) is a function and \(A\subseteq Y\), then \(f^{-1}(A)=\{x\in X: f(x)\in A\}\).