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Logic: Quantifiers
Some sentences feel an awful lot like statements but aren't. For example, n is even This is not a statement because it doesn't have a truth value; unless we know what n is, we can't really do much. Definition. A sente (More)
Logic: Conditionals
  So far our statements haven't been very interesting. In fact most mathematical statements of interest are things like If a function is differentiable, then it is continuous. These statements are known as conditional, because they depend o (More)
Logic: Or
Let's get another little word. Definition. Given two propositions P and Q, the proposition P\vee Q (read P or Q) has truth value as given below: P [l (More)
Logic: Not & And
Statements by themselves aren't that interesting; let's see how to combine them. "not" A writing teacher once told me the best way to clean up your writing is to eliminate as many adverbs and adjectives as possible. But one very important adverb--w (More)
Logic: Overview
Quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo dicere: Calculemus! When controversies arise, there will (More)
Cardinality
Consider the following question: Are there more Horsemen of the Apocalypse, or teams in the American Football Conference - East Division? The standard way to answer this question would be to count each collection. First let's do the Horsemen of the A (More)
Logic in Proofs
[callout headingicon="noicon" textalign="textright" url="https://www.youtube.com/watch?v=OncDlwNH9F0" target="true" type="basic"] If man is five, then the devil is six. And if the devil is six, then god is seven. This monkey's gone to heaven. Black F (More)
Logic: Uniqueness
There is another quantifier, besides existential and universal: the unique existential quantifier. Definition. The sentence $\exists!x:S(x)$ is true if there is exactly one $x$ in the universe so that $S(x)$ is true. We read There is a unique $x$ suc (More)
Logic: Proofs with Quantifiers
Now we're at the stage where you're probably antsy to start actually writing proofs. Excellent! Let's get started. [pullquote color="reynoldsred" align="alignright"]The logical form of a statement indicates the structure of its proof.[/pullquote] T (More)
The Natural Numbers and Induction
[callout headingicon="noicon" textalign="textright" type="basic"] Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk. Beloved God made the integers; everything else is the work of human hands. Leopold Kronecker [/callout] Mo (More)
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