# Propositional Logic

## Syntax and Symbols

SyntaxTerminologyExplanationExample
`P⇒Q`material implicationIf `P`, then `Q`.
Equivalent to `[~P ∨ Q]`.
`[(x=2)⇒(x²=4)]=⊤`,
but not `[(x²=4)⇒(x=2)]` as `x` could also be `−2`.
`P⇔Q`
`P≡Q`
material equivalence`Q`, if and only if `P`
Equivalent to `[(P⇒Q)∧(Q⇒P)]`.
`P` requires `Q` and `Q` requires `P`.
`~P`negationInverts the truth value of the statement.`~⊤ = ⊥`
`P∧Q`logical conjunction`P` and `Q`; True if both terms are true, false otherwise.`⊤∧⊤=⊤`; `⊤∧⊥=⊥`
`P∨Q`inclusive disjunctionTrue if either (or both) `P` or `Q` are true.`[⊥∨⊤]=⊤`; `[⊥∨⊥]=⊥`
`P⊕Q`
`P⊻Q`
exclusive disjunctionTrue if either (but not both) `P` or `Q` are true.
Equivalent to `[(PvQ)∧~(P∧Q)]`.
`⊤⊻⊤=⊥`
`⊤⊻⊥=⊤`
`⊥⊻⊥=⊥`
`⊤`tautologyAlways true.
`⊥`contradictionAlways false.
`∀x`universal quantificationFor any `x`
For each `x`
`∀x∈ℕ`
`x²>0`
For each `x` in the `ℕ`atural Numbers, `x²` is greater than zero.
`∃x`existential quantificationThere exists at least one `x`.`∃x:P(x)`
There is at least one `x` such that `P(x)` is true
`∃!x`uniqueness quantificationThere exists exactly one `x`.`∃!x:P(x)`
There is exactly one `x` such that `P(x)` is true
`x≔y`definition`x` is defined as `y`.
`P⊢Q`turnstile`P` proves `Q`.`(A→B)⊢(~B→~A)`

## Rules of Replacement

SyntaxTerminologyExplanation
`P∨Q≡Q∨P`
`P∧Q≡Q∧P`
commutativeSwap arguments for any `∨` or `∧`.
`P∨(Q∨R)≡(P∨Q)∨R`
`P∧(Q∧R)≡(P∧Q)∧R`
associativeParenthesis can be moved between `v` or `∧`.
`P∨⊥≡P`
`P∧⊤≡P`
identity`∨` with `⊥` becomes the non-`⊥` argument.
`∧` with `⊤` becomes the non-`⊤` argument.
`P∨~P≡⊤`
`P∧~P≡⊥`
negation`∨` with two opposite values must always be `⊤`.
`∧` with opposite values must always be `⊥`.
`~(~P)≡P`double negative`~` inverts values; `~~` inverts values back to initial value.
`P∨P≡P`
`P∧P≡P`
idempotent`∨` and `∧` operations evaporate when arguments are the same.
`P∨⊤≡⊤`
`P∧⊥≡⊥`
universal bound`∨` with `⊤` is always `⊤`.
`∧` with `⊥` is always `⊥`.
`~(P∨Q)≡~P∧~Q`
`~(P∧Q)≡~P∨~Q`
DeMorganThe `~` of an `∨` is the two terms negated and `∧`ed.
The `~` of an `∧` is the two terms negated and `∨`ed
`P∨(P∧Q)≡P`
`P∧(P∨Q)≡P`
absorption`∨` always requires `P`, so additional term evaporates.
`∧` always requires `P`, so additional term evaporates.
`~⊥≡⊤`
`~⊤≡⊥`
negation of tautologyTautology and contradictions are known values - each is equal to the other’s negations

## Set Symbols

SyntaxTerminologyExplanationExample
`A=B`equality`A` has the same elements as `B`.`{1,2,3}={1,2,3}`
`x∈A`element of`x` is in `A``A={1,2,3,4}⇒4∈A`
`x∉A`not element of`x` is not in `A`.`A={1,2,3,4}⇒5∉A`
`A⊆B`subset`A` is a included in `B` and/or `A=B`.`{1,3}⊆{1,2,3,4}` and `{1,3}⊆{1,3}`
`A⊊B`proper subset`A⊂B` and `A≠B`.`{1,3}⊊{1,2,3,4}` but NOT `{1,3}⊊{1,3}`
`A⊄B`not subset`A` is not a subset of `B`.`{1,2}⊄{1,4}`
`A∪B`unionThe result of adding `A` and `B`.`{1,2,3}∪{3,4}={1,2,3,4}`
`A∩B`intersectionThe common elements in `A` and `B`.`{1,2,3}∩{2,3,4}={2,3}`
`A\B`
`A-B`
relative complementThe elements from `A` not in `B`.`{1,2}\{1,3}={2}`
`A×B`cartesian productThe combinations resulting from pairing all elements in `A` with all elements in `B`.`{1,2}×{3,4}={(1,3),(1,4),(2,3),(2,4)}`
`{a⎮P(a)}`set builderA set of elements `a` with properties satisfying `P`.`{a∈ℤ⎮a≥0}=ℕ`
`Ac`
`A‘`
`Ā`
complementThe elements not in `A`.`Ac={x⎮x∉A}`
`A∆B`
`A⊖B`
symmetric differenceElements in either `A` or `B` but not both `A` and `B`.`{1,2,3}∆{2,3,4}={1,4}`
`⎮A⎮`
`#A`
cardinalityQuantity of values inside `A`.`A={1,2,3}`
`⎮A⎮=3`
`#A=3`

## Specifically Notated Sets

SymbolTerminologyExplanationExample
`∅`null setThe empty set`∅={}`
`𝕌`universal setall possible valuesanything `∈𝕌`
`ℕ₁`natural numberspositive counting numbers`ℕ₁={1,2,…}`
`ℕ`
`ℕ₀`
natural numberspositive counting numbers and zero`ℕ=ℕ₀={0,1,2,…}`
`ℤ`integersall discrete quantities including negative`ℤ={…,-2,-1,0,1,2,…}`
`ℚ`rational numbersfractions (including improper)`ℚ={a/b⎮a,b∈ℤ,~(b=0)}`
`ℂ`complex numbersnumbers formulated using `i``ℂ={a+bi⎮a,b∈ℝ,i²=-1}`
`5+3i∈ℂ`
`ℝ`real numbersall in quantities including negative`ℝ={x⎮-∞<x<∞}`
`π∈ℝ`