Solutions to Assignment 1, Part III

(a) [2 marks]
Both of those things may happen, but not at the same time
  EF Thing1 /\ EF Thing2 /\ ~EF(Think1 /\ Think2)

(b) [2 marks]
If it ever happens, it won't keep happening forever.
Perhaps it means that it won't keep happening continuously forever, in
which case the answer is AG AF ~that, or, if you prefer,
~EF EG that.  Or perhaps it means that after
a while it won't ever happen again, in which case the answer
is AF AG ~that, or, if you prefer, ~EG EF that

(c) [2 marks]  When this situation happens, it may persist for a while, but 
not forever, and is followed immediately by that situation
   A[this U ~this /\ that]

(d)[2 marks]  Whenever we get into that situation, we can get out
of it.  

If it means we can get out and stay out, then AF EG ~that.  
If it means we can get out but then we can get back in, then
AG EF ~that.

(2) [2 marks]
Whenever we get into that situation, we will get out of it.

If it means we will get out and stay out, then AF AG ~that.
If it means we will get out but we might get back in, then
AG AF ~that.

Which of the following formulas are equivalent?

(a) [3 marks]  EF p and EG p

  p
  |     models EF p but not EG p
  ~p

Note:  tihs tree diagram and all further tree diagrams continue downward
forever.  I show only the part at the top.

(b) [3 marks]  EF p \/ EF q   and EF(p \/ q)
equivalent

(c) [3 marks] AF p \/ AF q  and AF(p \/ q)
       ?
      / \              models AF(p \/ q) but not AFp \/ AF q
p /\ ~q  ~p /\ q

(d)[3 marks]  AF~p and ~EGp
equivalent

(e) [3 marks]  EF~p and ~AF p

     p
     |    models EF~p but not ~AF p
    ~p


(f) [4 marks]  A[p U A[q U s]] and A[A[p U q] U s]
     
     p /\ ~s /\ ~q
          |           models A[p U A[q U s]] but not A[A[p U q] U s]
     ~p /\ s /\ ~q