Represent the form of the following sentences in predicate logic using the = symbol where necessary.

There is one and only one Princeton University. (Use

*P**x*for “*x*is a Princeton University”)There is at most one Ivy League university in New Jersey. (Use

*I**x*for “*x*is an Ivy League university”, and use*N**x*for “*x*is in New Jersey.”)The smallest prime number is even. (

*P**x*,*E**x*,*x*<*y*, variables are restricted to numbers.)

Prove the following sequents using any of the rules, including the = intro and elim rules. You may write proofs in “sloppy mode”, i.e. you may combine steps, cut in results proved elsewhere, etc., as long as you explain clearly and convincingly how the proof works.

∃

*x*(*P**x*∧∀*y*(*P**y*→*x*=*y*)) ⊢ ∀*x*∀*y*((*P**x*∧*P**y*)→*x*=*y*)⊢ ∀

*x*∀*y*((*x*=*y*)→(*y*=*x*))

Let *R**x**y*
be a binary relation symbol that satisfies the transitivity axiom (page
126). Suppose that *R**x**y* satisfies two
other axioms: serial ∀*x*∃*y**R**x**y*
and irreflexive ∀*x*¬*R**x**x*.
Show that there are at least three distinct things. (Your proof need not
be fully formal, but it needs to be clear that you understand why the
moves you make are licensed by our system.)