程序代写代做代考 Logic Tutorial 2

Logic Tutorial 2

1. An Exercise taken from Year 1 Logic – with my thanks to Prof. Ian Hodkinson

Consider a set of objects labelled A, B, C, … placed on a 3*3 grid, and the following

atomic formulas talking about the objects:

[x next-to y] means (that is, it is true if) x and y are adjacent (horizontally or

vertically, but not diagonally);

[x sees y] means x and y are in the same row or the same column;

[x left-of y] means x is in a column to the left of the column of y;

[x above y] means x is in a row above the row of y.

(a) For the placements shown in figure below, which of the following evaluate to true, and

why?

i. [A sees B]  [B sees C]

ii. [B next-to D]  [B next-to E]

iii. ¬ ([A left-of F]  [F above A])
iv. ¬ ([E left-of D] → ¬[D next-to C]) → ¬[A sees E]

v. ([E sees D]  [F sees E]) → ¬([B above E]  [B next-to C])

A D

C F B

E

(b) Place the 6 objects A, … , F on the grid so that all the formulas above are true.

2. For each of the following determine if it is a tautology, inconsistency or contingency by drawing the

truth table.

a. P  (P  Q) d. (P  (Q  P))  P

b. (P  Q)  (P  Q) e. (P  Q)  ( P  Q)

c. Q   P  (P  (Q  P)) f. ((P  Q)  (R  S)  (P  R))  (Q  S)