1 Notational conventions

Here is a list of fairly standard concepts in mathematics that we will use in this module. You are mostly expected to be familiar with these concepts already.

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  The symbol := will mean “is defined to be”. Important new words will be in bold.

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  A set is a collection of distinct elements. If $A$ is a set, then the notation $x\in A$ means “$x$ is an element of $A$”.

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  $\mathbb{Z}:=\{\mathrm{\dots },-2,-1,0,1,2,\mathrm{\dots }\}$ is the set of integers.

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  $\mathbb{Q}:=\{{\displaystyle \frac{p}{q}}|p,q\in \mathbb{Z},q\ne 0\}$ is the set of rational numbers, or fractions. Here we are using set notation; in words it says “the collection of all numbers of the form ${\displaystyle \frac{p}{q}}$ where $p,q$ are both integers and $q$ is not 0.”

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  If $A$ and $B$ are sets, then $A\subset B$ means $A$ is a subset of $B$; in other words, every element of $A$ is also an element of $B$. This includes the case when $A=B$.

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  If $a,b\in ℝ$, then means that $b$ is strictly bigger than $a$, so it is not equal to $a$. The symbol $a\le b$ means that $b$ is bigger than or equal to $a$.

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  We will use logical quantifier symbols $\forall$ (“for all”) and $\exists$ (“there exists”).

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  $ℝ$ is the set of real numbers, including all the rational numbers and the irrational ones (such as $\pi ,e,\sqrt{2}$, etc…).

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  $ℂ$ is the set of complex numbers is $ℂ:=\{a+bi|a,b\in ℝ\}.$ Here the symbol $i$ denotes a square root of $-1$. So multiplication is defined by $(a+bi)(c+di):=(ac-db)+(ad+bc)i$.

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  The product of two numbers (or, more generally, two elements of a field) $a,b$ will be written as $ab$, or $a\cdot b$. For instance: $(-2)3=-2\cdot 3=-(2\cdot 3)=2\cdot (-3)=-6.$

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  A function $f$ from a set $A$ to a set $B$ will be written $f:A\to B$. This means that for every element $a\in A$, we assign an element in $B$, which we call $f(a)$. In other words, if $a\in A$ then $f(a)\in B$.