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Mathematical Induction
[callout headingicon="noicon" textalign="textright" type="basic"] 千里之行,始於足下 The journey of a thousand miles starts with a single step. Lao Tzu [/callout] Many statements in mathematics are true {\em for any natural number}. For example. Fundamental (More)
Complete Induction
[callout headingicon="noicon" textalign="textright" type="basic"] Travel isn't always pretty. It isn't always comfortable. Sometimes it hurts, it even breaks your heart. But that's okay. The journey changes you; it should change you. It leaves marks (More)
the Well-Ordering Principle
Here's a nice fact about the natural numbers: Well-Ordering Principle. Every nonempty collection of natural numbers has a least element. Observe, before we prove this, that a similar statement is not true of many sets of numbers. The interval $(0,1)$ (More)
[callout headingicon="noicon" textalign="textleft" type="basic"] I don't want to belong to any club that will accept people like me as a member. Groucho Marx [/callout] In our quest to reason about quantities and relationships, we'd like to handle q (More)
Set Operations and Subsets
Just like we defined logical formulae by giving truth tables, we can define set formulae by giving a criterion for membership. Definition. Given sets $A$, $B$, we define  the complement of $A$, $A^c$, by the rule $x\in A^c$ iff $x\notin A$.  th (More)
The Element's-Eye View
There are three things to keep in mind when it comes to set-theory proofs (which most of our proofs hereafter will be): Sets are defined by their membership. Because of this, we nearly always take an element's-eye view of any proof involving sets. (More)
Set-Builder Notation
Recall that, by definition, $S$ being a set is equivalent to ``$x\in S$" being an open sentence. This means that the theory of sets and the theory of logic are basically the same--differing only by notation and perspective. One can exploit this corre (More)
Families of Sets and `Generalized' Set Operations
Finite Set Operations The operations $\cup$ and $\cap$ apply to two sets; we can however use them to combine more than one set. If we have a finite collection of sets $A_1,A_2,\ldots,A_N$, we write \begin{equation}\begin{aligned}\bigcup_{i=1}^N A_i& (More)
[callout headingicon="noicon" textalign="textleft" type="basic"]Society does not consist of individuals, but expresses the sum of interrelations, the relations within which these individuals stand. Karl Marx, Grundrisse der Kritik der Politischen &Ou (More)
Basics of Relations
What's a relation? Definition.  A relation from the set $A$ to the set $B$ is a(ny) subset of $A\times B$. We call $A$ the source and $B$ the target or codomain of the relation. A relation from $A$ to $A$ is called a {\em relation on $A$}. If $R$ is (More)
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