Notation

This is a list of all the notation we will be using in this part of the course. It is all standard, and part of the purpose of this course is to familiarise you with symbols you will see in other courses but which might not be explained.

• \(\triangleq\): equal by definition (technically standard notation, but you may be more familiar with \(\equiv\) for this. But \(\equiv\) also means ‘equivalent to’).

• \(|v\rangle\): a vector.

• \(\left\{ a,b,c\right\}\): the set containing \(a,\)\(b,c\) (ordering doesn’t matter)

• \(\left(a,b,c\right)\): the sequence \(a,b,c\) (ordering matters)

• \(\left(a,b\right)\): an ordered pair (sequence of two elements)

• \(\forall\): for all

• \(\exists\): there exists

• \(|\): such that

• \(a\in S\): \(a\) is a member of the set \(S\)

• \(S\subset T\): \(S\) is a strict subset of the set \(T\)

• \(S\subseteq T\): \(S\) is a subset of the set \(T\) (i.e. it may be the same set)

• \(A\times B\): cartesian product

• \(f:S\times T\rightarrow U\): binary function \(f\) takes in an element of \(S\) and an element of \(T\) and returns an element of \(U\)

• \(f:S\times S\rightarrow S\): binary operation (binary function where all sets are the same set)

• \(a\implies b\): \(a\) implies \(b\)

• iff: if and only if

• \(a\Longleftrightarrow b\): \(a\) iff \(b\); equivalently, \(a\) implies \(b\) and \(b\) implies \(a\)

• \(\mathfrak{Re}(z)\): real part of \(z\)

• \(\mathfrak{Im}\left(z\right)\): imaginary part of \(z\)

• \(\aleph_{0}\) (aleph-null): the cardinality of the integers, a.k.a. countable infinity

• \(\mathfrak{c}\): the cardinality of the continuum