# Language proof and logic Chapter 15 question 21 how?

I'm really not understanding the set up of how to go about solving this problem any help is welcome

• What is the meaning of the empty screenshot ? – Mauro ALLEGRANZA Nov 29 '18 at 19:00

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