69 lines
1.6 KiB
Plaintext
69 lines
1.6 KiB
Plaintext
---
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format: markdown
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toc: yes
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title: Mathematical Proof Study Notes
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...
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# Texts
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- *Book of Proof*, Richard Hammack
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# Reading Notes
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## *Hammack*, 25 Jan 2014
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All of Mathematics can be described with *sets*.
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*set*
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: A collection of things. The things in the set are called *elements*.
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An example of a set: $\{2,4,6,8\}$
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The set of all integers:
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$$ \{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\} $$
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The dots mean the expressed pattern continues.
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Sets of infinitely many members are *infinite*, otherwise they are
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*finite*.
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Sets are *equal* if they have exactly the same elements.
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E.g. $\{2,4,5,8\} = \{4,2,8,6\}$ but $\{2,4,6,8\} \neq \{2,4,6,7\}$.
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Uppercase letters often denote sets, e.g. $A = \{1,2,3,4\}$.
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To express membership, we use $\in$, as in $2 \in A$.
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To express non-membership, we use $\notin$, as in $5,6 \notin A$.
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**Special Sets**
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$\Bbb{N}$
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: *natural numbers*, the positive whole numbers $\{1,2,3,4,5,\dots\}$
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$\Bbb{Z}$
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: *integers*, $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4,\dots\}$
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$\Bbb{Q}$
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: *rational numbers*, $\Bbb{Q} = \{x : x = \frac mn, m,n \in \Bbb{Z}
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and n \neq 0\}$.
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$\Bbb{R}$
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: *real numbers*, the set of all real numbers on the number line.
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$\emptyset$
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: *empty set*, the unique set with no members, $\{\}$
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* * * *
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For finite sets $X$, $|X|$ represents the *cardinality* or *size* of
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the set, which is the number of elements it has. E.g. $|A| = 4$.
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*set-builder notation* describes sets that are too big or complex to
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be listed out. E.g. the infinite set of even integers: $$
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E = \{2n : n \in \Bbb{Z}\}
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$$
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This can be read as "E is the set of all things of form $2n$, such
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that $n$ is an element of $\Bbb{Z}$."
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