For plomble #10: Since F is nonempty,suppose B ∈ F. Let x be an arbitrary element of B. Then since B ⊆ A,x ∈ A,so by the definition of ⋂F,x ∈ ⋂F.
Thus,since x was arbitrary,we have shown that x ∈ B implies x ∈ ⋂F,Therefore B ⊆ ⋂F.
#I think my proof is incorrect, Please help 🤗
hidden bone
versed yokeBOT
hidden bone
I think I jumped to conclusions by asserting x ∈ ⋂F after inferring x ∈ A,that’s another issue I felt I had here
I also think that supposing B ∈ F might’ve been a little sketchy?
versed yokeBOT
I think my proof is incorrect, Please help 🤗
gritty quarry
For plomble #10: Since F is nonempty,suppose B ∈ F. Let x be an arbitrary eleme...
"Then since B ⊆ A,x ∈ A,so by the definition of ⋂F,x ∈ ⋂F."
You should be explicit in invoking the premises. Here you need to state that since B is a subset of A for ALL A ∈ F, we have x ∈ A for ALL A ∈ F. This is what leads us to conclude x ∈ intersection(F)
hidden bone
Yes,that’s where I jumped to conclusions,I’ll edit it now 👍
Is there anything else about the proof you would fix?
gritty quarry
nope, the idea is fine
hidden bone
Oh,okay
hidden bone
"Then since B ⊆ A,x ∈ A,so by the definition of ⋂F,x ∈ ⋂F."
You should be explic...
Since F is nonempty,suppose B ∈ F. Let x be an arbitrary element of B. Then since B ⊆ A,x ∈ A,so since B is a subset of A for all A ∈ F, by definition of ⋂F,x ∈ ⋂F.
Thus,since x was arbitrary,we have shown that x ∈ B implies x ∈ ⋂F,Therefore B ⊆ ⋂F.
How’s this? Still too implicit?
gritty quarry
why are you assuming B ∈ F?
the only premise we have is
B is a subset of A for every A ∈ F
hidden bone
Yeah I don’t know either,that’s why I was sketchy
When I did the scratch work,I thought “Since F is nonempty,there is a set in F,so by existential instantiation,I’ll let B be such a set” I thought that maybe I could reach the conclusion that way by using the subset given,But since B was already mentioned as a set,that’s where I thought “Maybe that’s a little sketchy” How could we do it better?