Cheatsheet 01 - Notation & Logic
Mathematics for Master Students
Number Systems
\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\]
| Symbol | Name | Definition | Examples |
|---|---|---|---|
| \(\mathbb{N}\) | Natural numbers | \(\{1, 2, 3, \dots\}\) | \(1, 7, 42\) |
| \(\mathbb{Z}\) | Integers | \(\{\dots, -2, -1, 0, 1, 2, \dots\}\) | \(-5, 0, 3\) |
| \(\mathbb{Q}\) | Rational numbers | \(\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}\) | \(\frac{1}{3}, 0.75, -4\) |
| \(\mathbb{R}\) | Real numbers | All points on the number line | \(\sqrt{2}, \pi\), all of the above |
- In this tutorial \(0 \notin \mathbb{N}\): write \(\mathbb{N}_0 = \{0, 1, 2, \dots\}\) if zero is included.
- \(\sqrt{2}\) and \(\pi\) are irrational (real, but no fraction equals them): \(\pi \approx \frac{22}{7}\), but \(\pi \neq \frac{22}{7}\).
Intervals & Inequalities
| Notation | Inequality | Description |
|---|---|---|
| \([a, b]\) | \(a \leq x \leq b\) | Closed: both endpoints included |
| \((a, b)\) | \(a < x < b\) | Open: both endpoints excluded |
| \([a, b)\) | \(a \leq x < b\) | Half-open: \(a\) included, \(b\) excluded |
| \((a, b]\) | \(a < x \leq b\) | Half-open: \(a\) excluded, \(b\) included |
| \([a, \infty)\), \((-\infty, a]\) | \(x \geq a\), \(x \leq a\) | Unbounded: the \(\infty\) side is always open |
Absolute value: the distance of \(x\) from zero (\(|a - b|\) = distance between \(a\) and \(b\)):
\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \qquad\quad |x - a| \leq d \quad\Leftrightarrow\quad x \in [a - d,\, a + d]\]
Square bracket \([\) = endpoint included (it “grabs” it); round parenthesis \((\) = excluded. \(\infty\) always gets a parenthesis. It is not a number you could include.
Sum & Product Notation
\[\sum_{i=k}^{m} x_i = x_k + x_{k+1} + \dots + x_m \qquad\qquad \prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdot \dots \cdot x_n\]
Read \(\sum\) as a loop: for each index \(i\) from the lower limit \(k\) to the upper limit \(m\), add up the summand \(x_i\). The index name is arbitrary; \(\prod\) works the same, but multiplies.
\[\sum_{i=1}^{n} c \, x_i = c \sum_{i=1}^{n} x_i \qquad\quad \sum_{i=1}^{n} (x_i + y_i) = \sum_{i=1}^{n} x_i + \sum_{i=1}^{n} y_i \qquad\quad \sum_{i=1}^{n} c = n \cdot c\]
Double indices: \(x_{ij}\) = flow from origin \(i\) (first index, row) to destination \(j\) (second index, column).
- Leaving origin \(i\): \(\sum_{j=1}^{m} x_{ij}\)
- Reaching destination \(j\): \(\sum_{i=1}^{n} x_{ij}\)
- Total cost over all routes: \(\sum_{i=1}^{n} \sum_{j=1}^{m} c_{ij} \, x_{ij}\)
\(\sum_{i=k}^{m} x_i\) has \(m - k + 1\) terms. Both endpoints count. \(\sum_{i=3}^{10} x_i\) has \(8\) terms, not \(7\).
Greek Letters
| Letter | Name | Typical use | Letter | Name | Typical use |
|---|---|---|---|---|---|
| \(\alpha\) | alpha | significance level | \(\mu\) | mu | mean |
| \(\beta\) | beta | regression | \(\pi\) | pi | \(3.14159\dots\), profit |
| \(\delta, \Delta\) | delta | change, difference | \(\rho\) | rho | correlation |
| \(\varepsilon\) | epsilon | error term | \(\sigma, \Sigma\) | sigma | std. dev., sum |
| \(\lambda\) | lambda | arrival rate | \(\theta\) | theta | parameter |
Logic
A statement is either true or false, never both. Double negation cancels: \(\neg(\neg p) = p\). The “or” is inclusive; \(p \Rightarrow q\) is false only for the broken promise: \(p\) true, \(q\) false.
| \(p\) | \(q\) | \(\neg p\) | \(p \wedge q\) | \(p \vee q\) | \(p \Rightarrow q\) | \(p \Leftrightarrow q\) |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
De Morgan’s laws: negation flips “and” into “or”, and vice versa:
\[\neg(p \wedge q) \Leftrightarrow \neg p \vee \neg q \qquad\qquad \neg(p \vee q) \Leftrightarrow \neg p \wedge \neg q\]
Derived from \(p \Rightarrow q\): \(\quad p\) is sufficient for \(q\); \(\;q\) is necessary for \(p\).
| Derived statement | Form | Equivalent to \(p \Rightarrow q\)? |
|---|---|---|
| Contrapositive | \(\neg q \Rightarrow \neg p\) | ✓ |
| Converse | \(q \Rightarrow p\) | ✗ |
| Inverse | \(\neg p \Rightarrow \neg q\) | ✗ |
- Converse error: from \(p \Rightarrow q\) and \(q\), nothing follows about \(p\).
- The negation of “the warehouse is full” is “the warehouse is not full”, not “empty”.
- \(\Rightarrow\) and \(\Leftrightarrow\) are not interchangeable: \(x = 2 \Rightarrow x^2 = 4\) holds, but \(x^2 = 4 \Rightarrow x = 2\) fails (\(x = -2\)), so \(x^2 = 4 \Leftrightarrow x = 2\) is false.
Quantifiers
| Symbol | Meaning | Example |
|---|---|---|
| \(\forall\) | for all | \(\forall x \in \mathbb{R}: x^2 \geq 0\) |
| \(\exists\) | there exists | \(\exists n \in \mathbb{N}: n > 10^6\) |
Negation swaps the quantifier and negates the predicate. To disprove \(\forall\): one counterexample; to prove \(\exists\): one witness.
\[\neg\,(\forall x: P(x)) \;\Leftrightarrow\; \exists x: \neg P(x) \qquad\qquad \neg\,(\exists x: P(x)) \;\Leftrightarrow\; \forall x: \neg P(x)\]
“All shipments arrive on time” negates to “at least one shipment does not arrive on time”, not to “no shipment arrives on time”. One late shipment already refutes the claim.