231 lines
6.7 KiB
Plaintext
231 lines
6.7 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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* * * *
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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*, $\Bbb{Q} = \{x : x = \frac mn; m,n \in \Bbb{Z};
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n \neq 0\}$.
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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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*intervals*
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: For two numbers $a,b \in \Bbb{R}$ with $a < b$, we can form various
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intervals on the number line by listing them as a bracketed pair. A
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parenthesis indicates that side of the interval is *open*, while a
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square bracket indicates that side of the interval is *closed*. A
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closed interval *includes* the element of the pair on the closed side,
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while an open interval does not. Infinite intervals are denoted by
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including $\inf$ as one member of the pair on the open side.
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* * * *
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**Exercises 1.1**
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1. $\{4x-1 : x \in \Bbb{Z}\}$ is
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$\{\dots, -21, -16, -11, -6, -1, 4, 9, 14, 19,\dots \}$
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* * * *
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With two sets $A$ and $B$, one can "multiply" them to form the set $A
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\times B$ which is called the *Cartesian product*.
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**Definition 1.1**
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: An *ordered pair* is a list $(x,y)$ of two things $x$ and $y$,
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enclosed in parentheses and separated by a comma. They are
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distinguished by order; e.g. $(3,4) \neq (4,3)$.
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**Definition 1.2**
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: The *Cartesian product* of two sets $A$ and $B$ is another set, $A
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\times B$, defined as $A \times B = \{(a,b):a \in A, b \in B\}$.
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E.g. $\{0,1\} \times \{2,1\} = \{(0,2),(0,1),(1,2),(1,1)\}$
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**Fact 1.1**
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: If $A$ and $B$ are finite sets, then $|A \times B| = |A| \cdot |B|$.
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The set $\Bbb{R} \times \Bbb{R}$ can represent the points on the
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Cartesian plane.
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The idea extends to a 3-list, or *ordered triple*. In general:
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$$ A_1 \times A_2 \times \dots \times A_n = \{
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(x_1,x_2,\dots,x_n) : x_i \in A_i, i \in \Bbb{N} \}
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$$
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We can also take *Cartesian powers* of sets. For a set $A$ and
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positive integer $n$, $A^n$ is the Cartesian product of $A$ with
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itself $n$ times: $$
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A^n = A \times A \times \dots \times A = \{(x_1,x_2,\dots,x_n):x_i \in
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A,i \in \{1,\dots,n\}\} $$
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* * * *
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**Exercises 1.2**
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1. $A = \{1,2,3,4\}, B = \{a,c\}$
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a. $A \times B =
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\{(1,a),(2,a),(3,a),(4,a),(1,c),(2,c),(3,c),(4,c)\}$
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b. $B \times A =
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\{(a,1),(c,1),(a,2),(c,2),(a,3),(c,3),(a,4),(c,4)\}$
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d. $B \times B = \{(a,a),(a,c),(c,a),(c,c)\}$
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e. $\emptyset \times B = \emptyset$
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f. $(A \times B) \times B = \{((1,a),a),((2,a),a),
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((3,a),a),((4,a),a),((1,c),a),((2,c),a),((3,c),a),
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((4,c),a),((1,a),a),((2,a),c),((3,a),c),((4,a),c),
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((1,c),c),((2,c),c),((3,c),c),((4,c),c)\}$
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* * * *
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**Definition 1.3**
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: $A$ and $B$ are sets. If every element of $A$ is also an element of
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$B$, then $A$ is a *subset* of $B$ and we write $A \subseteq B$. If
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this is not the case, we write $A \not\subseteq B$, which means there is
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at least one element of $A$ that is not in $B$.
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**Fact 1.2**
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: It follows from **1.3** that for any set $B$, $\emptyset \subseteq
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B$. I.e., the empty set is a subset of every set.
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**Fact 1.3**
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: If a finite set has $n$ elements, it has $2^n$ subsets.
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This can be shown by drawing a decision tree starting with the empty
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set, with each fork representing a choice of whether to insert the
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next element of the set in question. Since there are two possibilities
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at each fork and $n$ elements to consider for insertion, that gives
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$2^n$ total leaves of the tree.
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* * * *
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**Exercises 1.3**
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1. Subsets of $\{1,2,3,4\}$: $\emptyset, \{1\}, \{2\}, \{3\}, \{4\},
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\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\},
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\{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}$
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* * * *
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**Definition 1.4**
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: $A$ is a set. The *power set* of $A$ is another set,
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$\mathscr{P}(A)$, defined to be the set of all subsets of
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$A$. I.e. $\mathscr{P}(A) = \{X:X\subseteq A\}$.
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**Fact 1.4**
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: If $A$ is a finite set, $|\mathscr{P}(A)| = 2^{|A|}$.
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* * * *
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**Definition 1.5**
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$A$ and $B$ are sets.
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- The *union* of $A$ and $B$ is the set $A\cup B = \{x : x \in A
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\text{ or } x \in B\}$
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- The *intersection* of $A$ and $B$ is the set $A\cap B = \{x : x \in
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A \text{ and } x \in B\}$
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- The *difference* of $A$ and $B$ is the set $A - B = \{x : x \in A
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\text{ and } x \notin B\}$
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The operations $\cup$ and $\cap$ obey the commutative law for sets,
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but $-$ does not.
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- If an expression involving sets uses only $\cap$ or $\cup$,
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parentheses are optional.
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- If it uses both $\cap$ and $\cup$, parentheses are required!
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* * * *
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We usually discuss sets in some context. Our sets in that context will
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naturally be subsets of some other set, which we call the *universal
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set* or just *universe*. If we don't know specifically which set it
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is, we call it $U$.
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For example, when discussing the set of prime numbers $P$, the
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*universal set* is $\Bbb{N}$. When we discuss geometric figures such
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as the set of points in a circle $C$, the universe would be
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$\Bbb{R}^2$.
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**Definition 1.6**
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: $A$ is a set in the universe $U$. The *complement* of $A$ or $\overline A$
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is the set $\overline A = U - A$.
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E.g. if $P$ is the set of prime numbers, then $$
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\overline P = \Bbb{N} - P = \{1,4,6,8,9,10,12,\dots\}
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$$ so $\overline P$ is the set of composite numbers and 1.
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**Definition 1.7**
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$A_1, A_2,\dots, A_n$ are sets.
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$$\begin{align}
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A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n &=
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\left\{x : x \in A_i
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\text{ for at least one set $A_i$, for }
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1 \leq i \leq n\right\}\\
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A_1 \cap A_2 \cap A_3 \cap \dots \cap A-n &=
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\left\{x : x \in A_i
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\text{ for every set $A_i$, for }
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1 \leq i \leq n \right\}\\
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\end{align}$$
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