Natural language is inherently ambiguous The same words in different contexts can mean different things.
- Ex ’OR’
- If-Then
| Proposition | not Proposition |
|---|---|
| A is B | DON’T/HELLO |
| If-Then | $a^2+b^2=c^2$ |
| $\exist a\exist b(a^2+b^2=25)$ |
A propositional variable is variable that is only allowed to be true OR false.
<aside> 💡 We like lower letters starting around p.(p, q, r, t, u, v)
T stands for true. F stands for false.
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<aside> ✏️ The negation of p is the statement ”It is not the case that p.”
We write this a $\neg p$ Also: $\sim p,\ \ \bar{p},\ \ p',\ \ Np,\ \ !p$
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Ex ~
When we’re lucky, we can simplify a negation linguistically.
p q $\neg P$ T T F T F F F T T F F T
<aside> 🤭 $p\wedge q$
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<aside> 😅 $p \vee q$
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| p | q | $p\wedge q$ | $p \vee q$ |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | F |
$$ p \oplus q \equiv q\oplus p\equiv \neg(p\lrarr q)\\ (p\oplus q)\oplus r\equiv p\oplus (q\oplus r)\\ p\wedge(q\oplus r) \equiv (p\wedge q)\oplus(p\wedge r) $$
| p | q | $p\oplus q$ |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |