Show your working and use indices where applicable.

1. This is a question about TFL. Attempt all parts of this question. (a) Explain what these three sentences mean, and explain the differences between them:

A C

A C

A C

(b) State and prove the Disjunctive Normal Form Theorem

(c) Explain what it means to say, of some connectives, that they are jointly expressively adequate. Show that '' and '' are jointly expressively adequate. You may rely upon your answer to part (b).

(d) Are the connectives '', '', '' and '' jointly expressively adequate? Explain your answer

2. Attempt all parts of this question. You must use the proof system from the course textbook, forallx: Cambridge Version.

(a) Show each of the following:

(i) (P Q) (Q P)

(ii) (P Q),P Q

(b) Show each of the following:

(i) x(Fx Gx) xFx xGx

(ii) x(Fx yRxy),x(Gx zRxz),x(wRxw (Fx Gx)) x(Fx Gx)

(iii) xyRxy, xy x = y yxRxy

3. Attempt all parts of this question.

(a) Using the following symbolization key domain: all physical objects Mx: x is a mug

Rx: x is red

Tx: x is a table

Bxy: x belongs to y

a: Alice

symbolize each of the following sentences as best you can in FOL. If any sentences are ambiguous, or cannot be symbolized very well in FOL, explain why

(i) Every mug belonging to Alice is red.

(ii) The table is red.

(iii) Alice's mug is red.

(iv) Alice's mug does not exist.

(v) Two mugs are on the table.

(vi) If the mug belongs to anyone, it belongs to Alice.

(vii) None of the mugs on the table is Alice's.

(viii) Every mug is on exactly one table, and on every table there is exactly one mug

(b) Show that each of the following claims iswrong. You may assume the conventionsfor representing interpretations described inthe coursetextbook, forallx: Cambridge Version.

(i) Fa,Ga, Fb,Gb,Fc,Gc x(Fx Gx)

(ii) x(Fx y(Gy Rxy z((Gz Rxz) y = z))) x(Gx y(Fy Ryx z((Fz Rzx) y = z)))

(iii) xyRyx,xRxx,xyRxy xy(x = y z(Rxz Ryz))

4. Attempt all parts of this question.

(a) Write down the axiom of extensionality. Then, using the standard notation, define the set-theoretic notions of: union, intersection, subset, proper subset and power set

(b) Give examples for each of the following:

(i) Three non-empty sets, A, B, and C, such that none of AB, BC and A C is empty, but such that (A B) C is empty

(ii) Two different non-empty sets, A and B, such that (A) (B) = (A B)

(c) Give examples for each of the following:

(i) a set whose intersection with its power set is not empty

(ii) a set whose intersection with the power set of its power set is not empty

(iii) a non-empty set that is a subset of the power set of one of its members.

(d) Write down the axioms of probability. Explain conditional probability.

(e) There are two equally probable hypotheses: either Bryce baked exactly 100 cupcakes today, or Bryce baked exactly 10 cupcakes today. In either case, Bryce piped unique numbers onto them: between 1 and 10, if there are exactly 10 cupcakes, or between 1 and 100, if there are exactly 100 cupcakes. Bryce hands you a cupcake, with the number piped onto it. How probable is it, now, that Bryce baked exactly 100 cupcakes today? Explain your reasoning, highlighting any assumptions that you have made.