(a) Either a is small or both c and d are large.

(b) If a is a small cube then b is a large tetrahedron.

(c) (SameSize(a, b) ^ SameSize(b, c)) –> Small(c)

(d) c is neither between a and b, nor in front of either of them.

(e) b is a cube only if a is too.

(f) a and b are the same shape just in case they are both cubes.

(g) a=b <–> SameSize(a, b)

(h) b is in the same column as c, which is in the same row as d, which in turn is in the same column as a.

(i) a=b is a necessary condition for a and b being in the same row.

(j) a=b is a sufficient condition for a and b being in the same row.

2.

(a) Prove (informally) the following: If B is a tautological consequence of A then A –> B is a tautology.

(b) Give an example to show that A –> B can be true even though B is not a tautological consequence of A.

(c) Use a truth table to determine whether the following is a tautology – explain your answer briefly but clearly:

3.

(a) Give an assignment of truth values to P, Q and R that makes the premises true and the conclusion false in the following argument:

| Q v R
| Q –> P
|–
| P

P =         Q =             R =

(b) What can you conclude about the validity of the argument?

4.
(a) Fill in the necessary justifications or missing steps to complete the following proof:

(b) Give an informal proof based on the following formal proof of:

| a is not a cube
|—
| a is not a small cube.

(c) Give a counter-example to show that the following argument is invalid.

(d) Since the argument in (c) is invalid, there is no Fitch proof of it. What is wrong with each of the steps of the following 'proof' that Fitch has marked as incorrect?

5. Complete truth tables for the following arguments. If the conclusion is a tautological consequence of the premise give a formal proof; if not either give a counter-example or explain why the argument is valid even so.

(a) (Tet(a) ^ Large(a)) v (Tet(a) ^ Small(a)) therefore Large(a) v Small(a)
 

(b)  (¬Adjoins(b,c) v RightOf(d,c)) ^ (¬RightOf(d,c) v Adjoins(c,b)) therefore ¬(¬Adjoins(b,c) ^ Adjoins(c,b)) — to save space in your truth table, you could let A(a,b) be short for Adjoins(a,b) and R(a,b) be short for RightOf(a,b)

Solutions

Back to Quiz page

Back to 102 Home Page