Appendix 1: Mathematical Logic

The purpose of this appendix is to give the reader a formal introduction to mathematical logic, emphasizing the skills and language that are needed to understand the arguments and proofs in this dissertation.

A mathematical expression is analogous to an English noun; it is a name given to some mathematical object of interest. A mathematical expression is often a number, a function, or a set.

A mathematical sentence expresses a complete thought. The next example explores the difference between expressions and sentences.

‘$\,(t_i,y_i)\,$’ is an expression. The chosen variables ($\,t_i\,$ and $\,y_i\,$) suggest that it represents an ordered pair of real numbers; a data point.

‘$\,\{(t_i,y_i)\}_{i=1}^N\,$’ is an expression. It is a set; a collection of data points.

‘$\,(t_i,y_i) = (i,i^2)\,$’ is a sentence. Note that sentences have verbs; the verb in this sentence is the equal sign, ‘$=$’.

‘IF data is generated by the rule $\,y_n = n\,,$ THEN $\,y_5 = 5\,$’ is a sentence. It is a sentence of the form ‘If $\,P\,,$ then $\,Q\,$’; sentences of this form are called implications, and will be treated in detail in this appendix.

Of particular importance in mathematics are sentences that are either true, or false, but not both, and these are called statements:

Capital letters, like $\,P\,$ and $\,Q\,,$ are frequently used to denote statements. Thus, $\,P\,$ could denote the true statement ‘$\,1+1 = 2\,$’, and $\,Q\,$ could denote the false statement ‘$\,1+1 = 3\,$’.

A variable is a symbol (often a letter) that is used to represent an arbitrary member of a specified set. This ‘specified set’ is called the universal set associated with the variable. Thus, the universal set gives the elements that one is allowed to draw on for a particular variable.

Many sentences become statements, once choices are made for the variables that appear in the sentence. For example, the truth of a sentence like ‘$\,x = 3\,$’ depends on the choice made for the number $\,x\,.$ If $\,x\,$ is $\,3\,,$ the sentence is true; otherwise, it is false. Such a sentence is called a conditional sentence:

The conditional sentence ‘$\,x = 3\,$’ can be conveniently denoted by $\,P(x)\,.$ Thus, $\,P(3)\,$ is true, but $\,P(y)\,$ is false for $\,y\ne 3\,.$

The conditional sentence ‘$\,x + y = 2\,$’ can be denoted by $\,P(x,y)\,.$ Thus, the sentences $\,P(1,1)\,$ and $\,P(0.5,1.5)\,$ are true, but $\,P(0,3)\,$ is false.

For simplicity, the following convention will be adopted for the remainder of this appendix: the notation $\,P(x)\,$ will be used to represent a conditional sentence, with the understanding that $\,x\,$ may represent more than one variable. For example, the conditional sentence $\,P(t,y)\,$ can be denoted by $\,P(x)\,,$ by defining $\,x = (t,y)\,.$

The sentence $\,\{1,2\} = \{2,1\}\,$ is a statement; it is true. Two sets are equal when they contain precisely the same elements.

The sentence $\,(1,2) = (2,1)\,$ is a statement; it is false. In order for two lists to be equal, corresponding entries must be equal. See Section 1.2 for properties of lists.

The sentence ‘The price of GE stock at 12:00 PM on 8/28/58 was greater than its price one day earlier at the same time’ is a statement. Even though the truth (true or false) of this sentence may be unknown to the reader, the sentence is either true, or false.

‘This sentence is false’ is not a statement. If it were true, then it would have to be false. If it were false, then it would have to be true. Therefore, it is neither true or false.

Connectives are used to make statements into ‘larger’ statements. The five basic connectives in mathematics are:

not,     and,     or,     $\implies\,,$     $\iff$

These connectives are defined via the truth table below, and are discussed in the following paragraphs:

The statement ‘$\text{not } P\,$’ is called the negation of $\,P\,.$ If $\,P\,$ is true, then ‘$\text{not } P\,$’ is false; if $\,P\,$ is false, then ‘$\text{not } P\,$’ is true.

The statement ‘$P \text{ and } Q\,$’ is true only when both $\,P\,$ and $\,Q\,$ are true. For example, the conditional sentence ‘$\,x = 1\,$ and $\,x = -1\,$’ is false for all real numbers $\,x\,,$ since a number cannot simultaneously equal $\,1\,$ and $\,-1\,.$

The statement ‘$P \text{ or } Q\,$’ is true when at least one of $\,P\,$ or $\,Q\,$ is true. For example, the conditional sentence ‘$\,x = 1\,$ or $\,x = -1\,$’ is true for $\,x \in \{1,-1\}\,,$ and false otherwise.

The reader is cautioned to distinguish between the English word ‘or’, and the mathematical word ‘or’. In English, the phrase ‘Carol or Bob went to the meeting’ usually means that either Carol went, or Bob went, but not both went. In mathematics, the statement ‘$P \text{ or } Q\,$’ is true if $\,P\,$ is true, or $\,Q\,$ is true, or both $\,P\,$ and $\,Q\,$ are true.

The statement ‘$P\implies Q\,$’ is read as ‘$P\,$ implies $\,Q\,$’, and is perhaps the most commonly-occuring type of mathematical sentence. Therefore, it should not be surprising that the sentence has many synonyms, including:

If $\,P\,,$ then $\,Q$
Whenever $\,P\,,$ then $\,Q$
$Q\,,$ if $\,P$
$\,Q\,,$ whenever $\,P$
$P\,$ is sufficient for $\,Q$
$Q\,$ is necessary for $\,P$

Given an implication $\,P\implies Q\,,$ the statement $\,P\,$ is called the hypothesis of the implication, and $\,Q\,$ is called the conclusion of the implication.

Roughly, a sentence of the form ‘$P\implies Q\,$’ is true if it has the property that whenever $\,P\,$ is true, then $\,Q\,$ is also true. If $\,P\,$ is true, but $\,Q\,$ is false, then the sentence ‘$P\implies Q\,$’ is false.

Observe from the truth table that if the hypothesis of ‘$P\implies Q\,$’ is false, then the statement ‘$P\implies Q\,$’ is true; in this case, ‘$P\implies Q\,$’ is said to be vacuously true. (The reason for this aspect of the definition should become clear, as soon as the discussion of statements of the form ‘For all $\,x\,,$ $\,P(x)\implies Q(x)\,$’ is completed.)

The sentence ‘$P\iff Q\,$’ is read as ‘$\,P\,$ is equivalent to $\,Q\,$’ or ‘$\,P\,$ if and only if $\,Q\,$’. The symbol ‘$\iff$’ is more frequently used when an equivalence is displayed (set off and centered); and the phrase ‘if and only if ’ is more frequently used when an equivalence appears in text.

The statement ‘$P\iff Q\,$’ is true exactly when $\,P\,$ and $\,Q\,$ have the same truth values; either they are both true, or they are both false.

Whenever two mathematical statements are equivalent, then they always have the same truth values, and hence can be used interchangeably. The next example presents an important mathematical equivalence.

The following truth tables prove that:

For all statements $\,P\,$ and $\,Q\,$:

These laws give the correct way to negate ‘and’ and ‘or’ statements, and are known as DeMorgan’s Laws.

Other equivalent statements, that are important for understanding the proofs in this dissertation, are presented later on in this appendix.

Many commonly-occurring mathematical sentences take the forms:

For all $\,x\,,$ $P(x) \implies Q(x)$
or
There exists $\,x\,$ such that $\,P(x)$
or
There exists a unique $\,x\,$ such that $\,P(x)$

The phrases ‘For all ’, ‘There exists ’, and ‘There exists a unique ’ are called mathematical quantifiers. Quantifiers, and quantified statements, are the next topic of discussion.

The sentence ‘$\,x = 3\,$’ has an implied universal set $\,{\cal U} = \Bbb R\,.$ The truth set is $\,\{3\}\,.$

The sentence ‘$\,\frac 1{|x|} = 2\,$’ has implied universal set $\,{\cal U} = \{x\in\Bbb R\ |\ x\ne 0\}\,.$ The truth set is $\,\{\frac 12,-\frac 12\}\,.$

The sentence ‘$\,x + y = 1\,$’ has implied universal set:

The truth set is $\,\{(x,1-x)\ |\ x\in\Bbb R\}\,.$ The graph of the equation $\,x + y = 1\,$ is a picture of its truth set; in this case, it is the line with $y$-intercept $\,(0,1)\,$ and slope $\,-1\,.$

Notice the use of the words ‘if and only if ’ in this definition. For example, in the second sentence, the words ‘if and only if’ are being used to compare the sentences:

For all $\,x\,,$ $\,P(x)$
and
the truth set of $\,P(x)\,$ equals $\,\cal U$

Therefore, these two sentences always have the same truth values. If one is true, so is the other; and if one is false, so is the other. Observe that if the sentence ‘the truth set of $\,P(x)\,$ equals $\,\cal U\,$’ is false, then there must exist $\,y\in\cal U\,$ for which $\,P(y)\,$ is false.

Thus, a counterexample is used to show that a mathematical sentence is not always true.

The quantified statement ‘For all $\,x\in\Bbb R\,,$ $\,\sqrt{x^2} = x\,$’ is false. For a counterexample, choose $\,y = -2\,.$ Then, the sentence ‘$\,\sqrt{(-2)^2} = -2\,$’ is false.

The quantified statement, ‘For all $\,x\ge 0\,,$ $\,\sqrt{x^2} = x\,$’ is true.

The quantified statement, ‘There exists a unique $\,x\in\Bbb R\,$ such that $\,x(x^2 + 1) = 0\,$’ is true. With universal set $\,\Bbb R\,,$ the truth set of $\,x(x^2 + 1) = 0\,$ is $\,\{0\}\,,$ which contains exactly one element.

The quantified statement, ‘There exists a unique $\,x\in\Bbb C\,$ such that $\,x(x^2 + 1) = 0\,$’ is false. Here, $\,\Bbb C\,$ denotes the set of complex numbers. With universal set $\,\Bbb C\,,$ the truth set of $\,x(x^2 + 1) = 0\,$ is $\,\{0,i, -i\}\,,$ which has more than one element.

Let $\,P\,$ be the quantified statement, ‘There exists a real number $\,x\,$ with $\,x^7 - \sqrt 2x^6 + 3x^4 - x + \pi = 0\,$’. Then, $\,P\,$ is true. It is not necessary to know what particular real number makes the conditional sentence ‘$\,x^7 - \sqrt 2x^6 + 3x^4 - x + \pi = 0\,$’ true; one need only establish that such a number does exist, and this is an easy consequence of the Intermediate Value Theorem.

To prove a quantified statement of the form

For all $\,x\,,$ $\,P(x) \implies Q(x)\,,$

it is necessary to show that the sentence ‘$\,P(x)\implies Q(x)\,$’ is true for every choice of $\,x\,$ from the universal set. If $\,P(x)\,$ is false, then the sentence ‘$\,P(x)\implies Q(x)\,$’ is vacuously true. Thus, one need only show that whenever $\,P(x)\,$ is true, so is $\,Q(x)\,.$ It is precisely for this reason that the sentence ‘$\,P(x)\implies Q(x)\,$’ is defined to be true, when $\,P\,$ is false.

The remainder of this appendix discusses the forms of proof that occur most frequently in this dissertation.

A direct proof of

For all $\,x\,,$ $\,P(x) \implies Q(x)\,,$

shows that whenever $\,P(x)\,$ is true, so is $\,Q(x)\,.$ For example, Lemmas $1$ and $3$ in Section 1.3 use direct proofs.

The statement ‘$\,\text{not }Q\implies \text{not } P\,$’ is called the contrapositive of the implication ‘$\,P\implies Q\,$’. The truth table below proves that an implication is equivalent to its contrapositive.

To prove a sentence of the form ‘$\,P\implies Q\,$’, one can equivalently prove ‘$\,\text{not } Q\implies \text{not }P\,$’, by a direct proof. Thus, the proof shows that whenever $\,Q\,$ is not true, then $\,P\,$ is not true.

For ease of notation, the symbols $\,\lnot\,$ for ‘not’, $\,\land\,$ for ‘and’, and $\,\lor\,$ for ‘or’ are introduced.

A contradiction is a sentence that is always false. For any statement $\,S\,,$ the sentence ‘$\,S\land(\lnot S)\,$’ is a contradiction.

Let $\,P\,,$ $\,Q\,$ and $\,S\,$ be statements. The logical equivalence

justifies the method of proof called proof by contradiction. To prove ‘$\,P\implies Q\,$’ by contradiction, one supposes that $\,P\,$ is true and $\,Q\,$ is false; and then reaches a contradiction.

For example, Lemma 4 in Section 1.3 uses a proof by contradiction.

To prove a sentence of the form ‘$\,P\implies (Q\lor R)\,$’, one often proves the logically equivalent sentence ‘$\,(P \land \lnot Q)\implies R\,$’ (see the truth table below).

For example, Theorem 2 in Section 1.4 uses this proof form.

To prove a sentence of the form ‘$\,(P\land Q)\implies R\,$’, one often proves the logically equivalent sentence ‘$\,(P\land \lnot R)\implies \lnot Q\,$’. The proof of this logical equivalence is left as an exercise.

To prove a sentence of the form ‘$\,P\iff Q\,$’, one usually proves the equivalent sentence ‘$\,(P\implies Q)\land (Q\implies P)\,$’. This logical equivalence justifies the use of the symbol ‘$\,\iff \,$’.

This form of proof is used for the Proposition in Section 1.3 .

The method of proof by induction is indispensible whenever it is desired to show that a statement $\,P(n)\,$ is true for all positive integers. The basic technique is:

• Show that $\,P(1)\,$ is true.
• Show that whenever $\,P(k)\,$ is true for a positive integer $\,k\,,$ then $\,P(k + 1)\,$ is also true.

If both of these steps can be accomplished, then since $\,P(1)\,$ is true, $\,P(2)\,$ must be true. And since $\,P(2)\,$ is true, then $\,P(3)\,$ must be true—and so on. This logic is sometimes referred to as the domino principle.

There are many variations on this technique. For examples, see Lemmas 2 and 5 in Section 1.3 .