Introduction to Mathematical Thinking, Week 3

続いて、Week 3、 Quantifier についてです。いわゆる、「∀(任意の) と ∃(存在する)」というやつですね。よく見かける。

講義自体は、YouTube で公開されてます。リンクがこちらにあるので、興味がある人はぜひ。
https://www.youtube.com/watch?v=LN7cCW1rSsI

※ あくまで素人が趣味で防備のノートとしてまとめているだけなので、解釈違い、数学的な厳密さの欠如についてはご容赦ください。

Lecture 5 – Quantifiers
SUPPLEMENT – How to Read Mathematical Formulas
その他色々
まとめ

Lecture 5 – Quantifiers

In mathematics, quantifiers are used to refer to the two extremes, there is at least one and for all.

∃(存在記号) とは、
There is an object x having a property P. ということであり、具体的には、

There is a real number x such that \( x^{2}+2x+1=0 \). のようになり、これは、

There exists a real number x such that \( x^{2}+2x+1=0 \). と同じ意味。

数学っぽく書くと、\( \exists x[x^{2}+2x+1=0] \) となる。

∃: Existential quantifier

\( \exists x[x^3+3x+1=0] \), so look at \( y=x^3+3x+1 \), which is a continuous function.

If x=-1then y=-3 and if x=+1 then y = 5. So the curve lies below the x-axis for x=-1, and above the x-axis for x=+1.

The point where the curve crosses the x-axis is where the x satisfies \( x^3+3x+1=0 \).

これを Indirect proof という。

A lot of mathematical proofs are of this nature. It shows that there is a solution to some equation or that there is an object that satisfies some property over there without finding such an object.

The same kind of argument I just used to show that a certain cubic equation has a real root can be used to prove the Wobbly Table Theorem.

初めて聞きましたが、Wobbly Table Problem：「ぐらつくテーブル問題」なるものがあるみたいですね。上で述べられている Indirect なアプローチで証明するみたいですが、今回はスルーしときます。また機会があれば眺めてみます。
https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/wobblytable/

Simply by rotating the table, you’ll be able to position it so that it doesn’t wobble.　ってことを示すみたいですね。

In fact, many mathematical statements that do not look like an existence statement on the surface turn out to be precisely that when you work out what they mean. For example, the statement that a number x is rational is an existence statement.

「\( \sqrt{2} \hspace{4mm} \text{is rational}\) 」は、結局のところ偽であるが。
So on the face of it, this doesn’t look like an existence statement. But it actually is, because we can write it as there exists natural numbers p and q such that \( \sqrt{2} = p/q\).

つまり、\( \sqrt{2} \hspace{4mm} \text{is rational}\) はすなわち、

There exist natural numbers p, q such that \( \sqrt{2} = p/q\).

数学っぽく書くと、 \( \exists p \exists q [\sqrt{2} = p/q]\) であり、これを否定することで、\( \sqrt{2} \) は、「有理数ではない」という結論を得る。

To make it more specific by writing it in the following way:

\( (\exists p \in \mathbb{N})(\exists q \in \mathbb{N})[\sqrt{2} = p/q] \)

where \( \mathbb{N} \) denotes the set of natural numbers.

\( (\exists p, q \in \mathbb{N})[\sqrt{2} = p/q] \)

If you start putting them together in this fashion there’s a possibility of getting confused and going off course. So I would say at the early stages, keep things distinct, as they are here, and try to avoid that for now.

For proving that \( \sqrt{2} \hspace{2mm} \) is not rational.

i.e. prove that \( \neg (\exists p \in \mathbb{N})(\exists q \in \mathbb{N})[2 = p^2/q^2] \)

One feature you need to get used to in mastering college mathematics, or more generally what I’m calling mathematical thinking, is the length of time you may need to spend puzzling about a problem or even one particular detail. とありますが、これは数学に限らず、ほとんどどんなものごとにでも当てはまりますね。要するに「時間を費やす」こと。

∀(全称記号) とは、
It asserts that something holds for all. という意味。

具体的には、\( \forall x \) means “for all x it’s the case that …”

“The square of any real number is greater than or equal to zero” は、以下のように書ける

\( \forall x (x^2 \geq 0) \) または \( (\forall x \in \mathbb{R} ) (x^2 \geq 0) \)

∀: Universal quantifier

quantifiers は、組み合わせて使われることも多い。

“There is no largest natural number” は、\( (\forall m \in \mathbb{N})(\exists n \in \mathbb{N})(n \gt m) \) と表現する。「全ての自然数 m において、n > m となる自然数 n が存在する」

Note that the order of the quantifiers is important.

もし仮に、\( (\exists n \in \mathbb{N})(\forall m \in \mathbb{N})(n \gt m) \) と記述した場合、「自然数 n が存在し、全ての自然数 m について n > m となる」ことを意味する。

The result says there is a natural number n which has the property that for all natural numbers m, m is bigger than n. In other words, this says there is a natural number bigger than all natural numbers, which is FALSE.

同様に、The American Melanoma Foundation による、”One American dies of melanoma almost every hour.” という英文は、

\( \exists A \forall H \text{[A dies in hour H]} \) となり、その意味は、「あるアメリカ人がいて、そのアメリカ人は1時間に1回死亡する。」という意図しない意味となる。

What it meant was:
\( \forall H \exists A \text{[A dies in hour H]} \) が正しい表現。

It will be different Americans for different hours, now as I mentioned earlier In the case of everyday English, these are almost never a problem. Everyone understands the context, we know what’s meant.

SUPPLEMENT – How to Read Mathematical Formulas

Binding precedence

Quantifiers bind whatever comes next.
\( (\forall L)(\cdots) \hspace{10mm} (\exists L)(\cdots)\)
So does negation.
\(\lnot(\cdots) \)
Next priority is conjunction.
\( (\cdots) \land (\cdots)\)
Then come disjunction and implication (conditional)
\( (\cdots) \lor (\cdots) \hspace{10mm} (\cdots) \Rightarrow (\cdots) \)

\( (\forall x) [P(x) \Rightarrow Q(x)] \)
For every x, if P(x) then Q(x). -> A strong statement, which we’ll frequently see.

\( (\forall x) [P(x) \land Q(x)] \)
For every x, P(x) and Q(x). -> A somewhat “strong” statement.

\( (\exists x) [P(x) \land Q(x)] \)
There is an x for which P(x) and Q(x). -> A strong statement, which we’ll frequently see.

\( (\exists x) [P(x) \Rightarrow Q(x)] \)
There is an x such that if P(x) then Q(x). -> A weak statement, which is fairly rare to see.

その他色々

In fact originally zero wasn’t a number it was just a zero symbol, which was a circle that denoted there was nothing there, and you needed to denote that there was nothing there when you had place value arithmetic.
* Place value is the value of each digit that appears in a number.

Sometimes we use sets, sometimes we use predicates. Sometimes we’ll use quantify restrictions, say for example, \( (\forall x \in \mathbb{R}) \). Sometimes we use predicate restrictions, say, \( Q(x) \).

Uniqueness quantification: \( \exists ! x[P(x)] \)
which is logically equivalent to
\( \quad \exists x [P(x) \land \forall y [P(y) \Rightarrow x=y]] \)
and equivalent to
\( \quad \exists x [P(x) \land \lnot \exists y [P(y) \land x \neq y]]\)

参考：https://en.wikipedia.org/wiki/Uniqueness_quantification

「\( \lnot \forall \)」は「\( \exists \) と \( \lnot \) の組み合わせ」に言い換えられる。
イメージは、『「すべて」の否定』は「すべてではない」のではなく、「すべてがそうではない」ということ、換言すると「ひとつも〜ない」ということ。