Answer : All are Satisfiable and None is Valid.
A First-order logic(FOL) formula $F$ is said to be Satisfiable if for some interpretation it is True. And a FOL $F$ is said to be Valid if for ALL interpretations it is True.
An Interpretation of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set.
Statements from $1\,\,\,to\,\,\,6\,:$
Now coming to the FOL statements from $1\,\,to\,\,\,6$, they are All Satisfiable. i.e. for some interpretation, they can be made True.
Let me show you one such interpretation :
Let the Domain $D$ consists three elements $\{ a,b,c \}$ (This is our Domain of discourse or Universe) and Let the predicate $J$ value for each pair of elements in the Domain be $J(x,y) = F, $ $\forall x,y \in D$
For this interpretation,
$A = (∀x)(∃!y)J(x,y)$ will become False and $B = (∃!y)(∀x)J(x,y)$ will become False. So, $A \equiv B $ will become True and $A \rightarrow B$ will become True. So, Statements $1,3,5$ will become True for this interpretation. (NOTE that in logic, operator $\equiv$ and $\leftrightarrow$ are one and same)
$C = (∃!x)(∀y)J(x,y)$ will become False and $D = (∃!y)(∀x)J(x,y)$ will become False. So, $C \equiv D $ will become True and $C \rightarrow D$ will become True. So, Statements $2,4,6$ will become True for this interpretation.
For this interpretation, Each FOL statement from $1$ to $6$ will become True, Hence, they are all Satisfiable.
Statements from $7\,\,\,to\,\,\,12\,:$
Now, If we find at least one Interpretation for which these statements (from 7 to 12) become False, we can say that they are Not Valid.
So, for that, Let me show you one such interpretation for statements $7,9,11$:
Let the Domain $D$ consists three elements $\{ a,b,c \}$ (This is our Domain of discourse or Universe) and Let the predicate $J$ value for each pair of elements in the Domain be $J(x,x) = T, $ $\forall x \in D$ and $J(x,y) = F,$ where $x \neq y$
For this interpretation,FOL statements $7,9,11$ will become False, Hence, they are Not Valid.
Now,
Let me show you one such interpretation for statements $8,10,12$:
Let the Domain $D$ consists three elements $\{ a,b,c \}$ (This is our Domain of discourse or Universe) and Let the predicate $J$ value for each pair of elements in the Domain be $J(a,x) = T, $ $\forall x \in D$ and $J(x,y) = F,$ where $x \neq a$
For this interpretation,FOL statements $8,10,12$ will become False, Hence, they are Not Valid.
