Basic set & number theory
This section corresponds to Basic set/number theory — 5:19 in the video Proof of Fermat Last Theorem FROM SCRATCH. Noah Jang is the original creator of the exposition and examples referenced here. The interactive implementations, animations, additional explanations, and contextual material are my own work.
🏷️ Common number sets
We will use the following standard number systems:
👉 \(\mathbb{N}\): the set of natural numbers
\[ \mathbb{N} = \{1, 2, 3,\cdots\} \]
👉 \(\mathbb{Z}\): the set of integers. Nice to know: the letter “Z” originates from the German word for numbers, namely “Zahle”.
\[ \mathbb{Z} = \{\cdots, -2, -1, 0,1, 2, 3,\cdots\} \]
👉 \(\mathbb{Q}\): the set of rational numbers
\[ \mathbb{Q} = \left\{\frac{a}{b}\ \bigg|\ a, b \in \mathbb{Z}, b \neq 0\right\} \] 👉 \(\mathbb{R}\): the set of real numbers
👉 \(\mathbb{C}\): the set of complex numbers
\[ \mathbb{C} = \{x +iy\ |\ x, y \in \mathbb{R}\} \]
🏷️ Carthensian product
For sets \(A\) and \(B\), the Cartesian product is defined as
\[ A \times B := \{(a,b)\ |\ a \in A, b\in B \} \]
Note that it is the “\(\times\)”-symbol that specifically defines how a new set is formed from two other sets.
✍️ Example 1
Let \(A=\{ 1, 2\}\) and \(B=\{3, 4, 5\}\), then
\[ A \times B = \{(1,3), (1,4), (1,5), (2,3), (2,4), (2,5) \} \]
✍️ Example 2
Let \(A=\{ 1, 2\}\), then
\[ A \times A = A^2 = \{(1,1), (1,2), (2,1), (2,2) \} \]
✍️ Example 3: \(\mathbb{R}^2\)
\[ \mathbb{R}^2= \left\{ \begin{pmatrix} x \\ y \end{pmatrix}\ \bigg|\ x,y \in \mathbb{R}^2 \right\} \]
🏷️ Symbols and abbreviations
- \(\forall\): for all
- \(\exists\): there exists
- \(:=\): is defined as
- \(w.l.o.g\): without loss of generality
- \(s.t.\): such that
- \(WTS\): want to show
- \(ETS\): enough to show
🏷️ Divisibility and congruence
- \(a|b\) means \(a\) divides \(b\), i.e., there exists \(k\) such that \(b=ak\).
- \(a\nmid b\) means \(a\) does not divide \(b\).
- For integers \(a, b\), and \(n\geq 1\),
\[ a \equiv b\quad (\text{mod}\ {n})\quad \text{means}\ n\ |\ (a-b). \]
✍️ Examples
\[ 17 \equiv 32\quad (\text{mod}\ 5), 15 \equiv 29\quad (\text{mod}\ 7) \]
🏷️ Greatest common divisor
For integers \(a,b\) the greatest common divisor of \(a\) and \(b\) is denoted by \(\gcd(a,b)\). Another common way to denote it is \((a,b)\).
- If \((a,c) = 1\), then \((ab,c)=(b,c)\)
- For any integer \(k\), \((a,b)=(a,b-ak)\)
- If \((a,b) = 1\) and \(a\ |\ bc\), then \(a\ |\ c\)
🚨 Why do we need this?
The above three properties are used to proof Fermat’s last theorem for the cases \(n=3\) and \(n=4\). Since these cases won’t be included on this site, we will skip the proofs and details.
✍️ Examples
- \(\gcd(24, 36)=12\)
- \(\gcd(28, 49)=7\)
- \(\gcd(17, 32)=1\) (co-primes)