User talk:JoshuaDaugherty

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Cheers! prime mover (talk) 05:45, 13 July 2021 (UTC)

P=NP

 * Theorem:

Consider the set of sets P such that P is the empty set ∅

Then the intersection of P is NP:

P = ∅ ⟹ ⋂P = NP

where NP is the universe

_________________________

Intersection of Empty Class, in the same form:

Let NP be a basic universe

Let ∅ denote the empty class

Then the intersection of ∅ is NP:

⋂∅ = NP

_________________________


 * Proof:

Let P = ∅

Then from the definition:

⋂P = {x : ∀X ∈ P : x ∈ X}

Consider any x ∈ NP

Then as P = ∅, it follows that:

∀X ∈ P : x ∈ X

from the definition of vacuous truth

∀x : P(x) ⟹ NP(x)

It follows directly that:

⋂P = {x : x ∈ NP}

That is:

⋂P = NP

The empty set contains itself and therefore contains 1

∀X : X ⊆ ∅ ⇒ X ⇒ ∅

It then follows that if P = ∅ = 1 and NP = 1

That is since 1 = 1

Then likely:

P = NP