via VIZ

via enchantedconsole

We’ve started work on some histograms of the orbits under the logistic family. Code and more pictures at viz.

via ffffound

Super nice video of the Hopf Fibration.

via flowingdata

via zerohedge – Filed Under Exponential

BT vs USD over 1 month via bitcoincharts

Over the last few months, the price of a bitcoin in USD blew up 6400% from $.50 to $32 before dropping as low as $15 over the last 2 days. This extreme market activity (along with the sensationalization of Silk Road) has caused a rapid increase in bitcoin investment and commentary. As a bitcoin enthusiast, I’ve decided to join the ranks of pro-bitcoin bloggers. This is at least partially a response to the lamestream meadia’s misunderstanding of and perhaps even hostility towards bitcoin. Keep reading for some analysis… Continue reading »

So far, we’ve introduced sets as the foundation of mathematics and we’ve shown how to construct the natural numbers as power sets of the empty set. I don’t think we can go much further without introducing functions. The use of functions is ubiquitous in mathematics. When most people think about functions, they think of functions of real numbers and picture graphs like this:

However, the idea of a function is foundational and far precedes the construction of the real numbers in formal mathematics.

In order to precisely define functions, we need to talk a little about products and relations between sets. The product of two sets X and Y is the set consisting of ordered pairs of elements from X and Y:

X [pmath size=12]*[/pmath] Y = {(x,y) such that x [pmath size=12]in[/pmath] X and y [pmath size=12]in[/pmath] Y}

For example, {1,2} [pmath]*[/pmath] {a,b,c} = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}. A relation R between two sets X,Y is just a subset of X [pmath]*[/pmath] Y:

[pmath]R subset X * Y[/pmath]

A relation is called left total if for every x [pmath] in [/pmath] X there exists y [pmath] in [/pmath] Y such that (x,y) [pmath] in [/pmath] R. A relation is called functional if for all [pmath] x in X[/pmath], [pmath] y,z in Y [/pmath] then y=z whenever [pmath](x,z) in R[/pmath] and [pmath](x,y) in R[/pmath]. A function is a relation that is both functional and left total. Ie. A function is a relation that relates a unique y [pmath] in [/pmath] Y to each x [pmath]in[/pmath] X. We generally denote a function “f” from a source set X to a target set Y by:

[pmath]f: X right Y[/pmath]

We often use graphs to visualize functions. For example, the following image represents a function “f” from the set {0, 1, 2, 3, … , 8} to itself:

In coming posts we’ll use functions to describe the cardinality of sets, as well as the addition and multiplication of natural numbers. Together, sets and functions are the basic tools of most mathematics.

Prev: Math II: Natural Numbers

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