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title: Mathematical Proof Study Notes
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# Texts

- *Book of Proof*, Richard Hammack

# Reading Notes

## *Hammack*, 25 Jan 2014

All of Mathematics can be described with *sets*.

*set*
: A collection of things. The things in the set are called *elements*.

An example of a set: $\{2,4,6,8\}$

The set of all integers:
$$ \{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\} $$

The dots mean the expressed pattern continues.

Sets of infinitely many members are *infinite*, otherwise they are
*finite*.

Sets are *equal* if they have exactly the same elements.

E.g. $\{2,4,5,8\} = \{4,2,8,6\}$ but $\{2,4,6,8\} \neq \{2,4,6,7\}$.

Uppercase letters often denote sets, e.g. $A = \{1,2,3,4\}$.

To express membership, we use $\in$, as in $2 \in A$.

To express non-membership, we use $\notin$, as in $5,6 \notin A$.

**Special Sets**

$\Bbb{N}$
: *natural numbers*, the positive whole numbers $\{1,2,3,4,5,\dots\}$

$\Bbb{Z}$
: *integers*, $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4,\dots\}$

$\Bbb{Q}$
: *rational numbers*, $\Bbb{Q} = \{x : x = \frac mn, m,n \in \Bbb{Z}
and n \neq 0\}$.

$\Bbb{R}$
: *real numbers*, the set of all real numbers on the number line.

$\emptyset$
: *empty set*, the unique set with no members, $\{\}$

* * * *

For finite sets $X$, $|X|$ represents the *cardinality* or *size* of
the set, which is the number of elements it has. E.g. $|A| = 4$.

*set-builder notation* describes sets that are too big or complex to
 be listed out. E.g. the infinite set of even integers: $$
 E = \{2n : n \in \Bbb{Z}\}
 $$
This can be read as "E is the set of all things of form $2n$, such
 that $n$ is an element of $\Bbb{Z}$."