Lecture from: 18.09.2024 | Video: Videos ETHZ
What is Discrete Mathematics?
Discrete maths works with finite or countably infinite mathematical structures. In the following lectures we’ll be looking at the following three parts:
- Discrete Structures
- Abstraction and Generalisation
- Mathematical Derivation
Mathematical Statements
In language and real life, statements are often dependent on their context and are subjective. In maths however statements are strictly true or false, independent of subjectiveness or context.
In maths whenever you see a statement ask yourself:
- It a well defined statement, does it make sense?
- Is it true or wrong
- Why is that the case
Example: 91 is a prime number:
- Is it a well defined statement? Yes
- Is it true? No
- Why? Because: 91 = 7 x 13
Composition of Statements
We can make new mathematical statements (which as mention above are either true or false) can be constructed out of other statements. This can be done in the following ways:
- Negation: A is false.
- And: A and B are both true.
- Or: A least one of A or B is true.
- Implication: If A is true, then B is true. Denoted by:
Implication
Implication might seem confusing in the beginning but it need not be. An implication is only false if A is true but B is false. It can be thought of like breaking a promise.
Say I were to tell you: “If I teach you, you’ll pass.”. Then there are 4 possibilities.
| I teach you | You pass | Am telling the truth (i.e. not lying)? |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T (you might pass regardless, doesn’t mean I lied) |
| F | F | T (you might also not pass, doesn’t mean I lied) |
Continue here: 02 Propositional Logic and Formulas