> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/310-mathematical-logic/truth-tables.md).

# Truth tables

A *truth function* is a function whose domain and codomain is a Boolean set,\
𝔹 = {true, false}. True and false are sometimes referred to as *truth values*.

That is, unary truth functions take a Boolean to a Boolean, 𝔹 -> 𝔹, while polyadic truth functions take an n-tuple of Booleans to a Boolean. In fact, all truth functions always return a single Boolean value.

taking a n-tuple of Booleans (truth values) and returns a single truth value, i.e. a single Boolean value (True or False). All truth function always return a single Boolean.

* A unary truth function takes a Boolean (1 var)    

  `Bool -> Bool`
* A binary truth function takes a 2-tuple of Booleans (2 vars)    

  `(Bool, Bool) -> Bool`
* A binary truth function takes a 3-tuple of Booleans (3 vars)    

  `(Bool, Bool, Bool) -> Bool`

and so on.

where:

* v is number of varibles
* c is number of configurations
* f is number of truth functions

If we let single letters variables (e.g. p, q, r, s, t, etc.) stand for values of Boolean type (each one is either T or F), we see that

the number of truth functions is related to the number of variables, n, in this way: (2^2)^n, which is 4^n. One Boolean variable ranges over 2 values so there are 2^m possible configurations:

v is num of configurations: c = 2^v

| v  | c  | f   | arity      |                                    |
| -- | -- | --- | ---------- | ---------------------------------- |
| 0  | 2  | 2   | nullary    | constants TRUE and FALSE, α -> 𝔹  |
| 1  | 2  | 4   | unary      | p: 𝔹 -> 𝔹                        |
| 2  | 4  | 8   | binary     | ⟨p: 𝔹, q: 𝔹⟩ -> 𝔹               |
| 3  | 8  | 64  | ternary    | ⟨p: 𝔹, q: 𝔹, r: 𝔹⟩ -> 𝔹        |
| 4  | 16 | 256 | quaternary | ⟨p: 𝔹, q: 𝔹, r: 𝔹, s: 𝔹⟩ -> 𝔹 |
| 5  | 25 | 256 | quinary    |                                    |
| 6  |    |     | senary     |                                    |
| 7  |    |     | septenary  |                                    |
| 8  |    |     | octonary   |                                    |
| 9  |    |     | novenary   |                                    |
| 10 |    |     | denary     |                                    |
| 11 |    |     | undenary   |                                    |
| 12 |    |     | duodenary  |                                    |
| 12 |    |     | tredenary  |                                    |

quaterdenary

An atomic logic variable, e.g. `p`, can have one of 2 possible values: it can either be true (T, 1) of false (F, 0). A truth function is an n-ary function from n variables to a truth value (T or F).

For any number n, there are 2^2^n possible n-ary truth functions.

## One variable

¬ ∧ ∨ → ← ⇔

p p p ταυ ταυ ταυ p ¬p p∧p p∨p p→p p←p p⇔p 1 0 1 1 1 1 1 0 1 0 0 1 1 1

## Two variables

← → ≡ ⇔ ⇒ ⊢

```
   |q←p|p←q|
```

| p q | ∧ | ∨ | p→q | q→p | p→(p→q) | p→(q→p) |                                                |
| --- | - | - | --- | --- | ------- | ------- | ---------------------------------------------- |
| 1 1 | 1 | 1 | 1   | 1   | 1  1 1  | 1  1 1  | .                                              |
| 1 0 | 0 | 1 | 0   | 1   | 1  0 0  | 1  0 1  | .                                              |
| 0 1 | 0 | 1 | 1   | 0   | 0  1 1  | 0  1 0  | .                                              |
| 0 0 | 0 | 0 | 1   | 1   | 0  1 1  | 0  1 1  | .                                              |
| --- | - | - | --- | --- | ------- | ------- | ---------------------------------------------- |

c a 8 e b d

1000 8 1010 a 1100 c 1110 e

## Three variables

─┬─┬─┬───┐0123456789abcdef │ │ │ ├───────────────────────────────────────────────────────────────────── n│4│3│ 2 │ ─├─┼─┼───┤ x│d│c│a b│F∧ ∨T ─┼─┼─┼───┼────────┬──────────────────────────────────────────────────────────── 0│0│0│0 0│00000000│11111111 1│0│0│0 1│00001111│00001111 2│0│0│1 0│00110011│00110011 3│0│0│1 1│01010101│01010101 ─┤ │ └───┼────────┴──────────────────────────────────────────────────────────── 4│0│1 0 0│ 5│0│1 0 1│ 6│0│1 1 0│ 7│0│1 1 1│ 8│1│0 0 0│ ─┤ └─────┼───────────────────────────────────────────────────────────────────── 9│1 0 0 1│ a│1 0 1 0│ b│1 0 1 1│ c│1 1 0 0│ d│1 1 0 1│ e│1 1 1 0│ f│1 1 1 1│ ─┴───────┴─────────────────────────────────────────────────────────────────────

## Misc ternary formulas

(p -> q -> r) -> (p -> q) -> p -> r

p ⇒ (q --> r) ⟾ (p ⟶ q) ⇒ (p ⟶ r) 0 0 0 1\
0 0 1 1\
0 1 0 0 1\
0 1 1 1\
1 0 0 1 0 0\
1 0 1 1 0\
1 0 1 0 0 1 0 0\
1 1 1 1\
≡ τ ≡


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