I'm really not understanding the set up of how to go about solving this problem any help is welcome
I'm really not understanding the set up of how to go about solving this problem any help is welcome .
Let us have a look. You have a definition of union of sets, and a definition of equality for sets. You need to show two unions are equal for some free terms a and b.
| [a,b] Free Terms (implicit)
| Ɐx Ɐy Ɐz (z ϵ Union(x,y) ↔ (z ϵ x ˅ z ϵ y)) Premise I (defines Union)
|_ Ɐx Ɐy (Ɐz (z ϵ x ↔ z ϵ y) → x = y) Premise II
| : :
| : :
| Union(a,b) = Union(b,a) Magic?
Okay... The first step is obviously to use Universal Elimination to instantiate to some terms.
Tip: If a and b are both terms, then Bivariate Functions Union(a,b) and Union(b,a) are also terms.
| [a,b] Free Terms
| Ɐx Ɐy Ɐz (z ϵ Union(x,y) ↔ (z ϵ x ˅ z ϵ y)) Premise I
|_ Ɐx Ɐy (Ɐz (z ϵ x ↔ z ϵ y) → x = y) Premise II
| Ɐy (Ɐz (z ϵ Union(a,b) ↔ z ϵ y) → Union(a,b) = y) Ɐ Elimination (of Premise II)
| Ɐz (z ϵ Union(a,b) ↔ z ϵ Union(b,a)) → Union(a,b) = Union(b,a) Ɐ Elimination (of above)
| : :
| : :
| Union(a,b) = Union(b,a) → Elimination ?
So... There you go. Take it away.
Using some of what was previously shown above I was able to to make a similar outline and add a subproof to figure out the answer
