(Pts = points, W = wins, D = draws, L = losses, GF = goals for, GA = goals against,
GD = goal differential). Flag Country Pts W D L GF GA GD. Chile. 3. 1. 0 1. 5. 3.
SOLUTIONS 1
1. After applying rules (a)-(c) we have a three way tie with Chile, Morocco, and Turkey. According to the regulations we need to look at the games played between those three teams to break the tie. The relevant results appear below. (Pts = points, W = wins, D = draws, L = losses, GF = goals for, GA = goals against, GD = goal differential)
Flag
Country
Pts
W D L
GF
GA GD
Chile
3
1
0
1
5
3
+2
Morocco
3
1
0
1
2
3
-1
Turkey
3
1
0
1
3
4
-1
Morocco 2 Turkey 0
Chile 3 Morocco 0
Turkey 3 Chile 2
Within rules (d) - (f) there is mention of the teams concerned. We will make explicit reference to this term throughout the answers below to clarify the set of teams to which the rules are being applied. a) Justify why Chile advances to the second round. [5 marks] Let us start our reasoning from rule (d) with the teams concerned being Chile, Turkey, and Morocco. Applying rule (d) does not break the tie since each team has three points (one win and one loss) against each other. Therefore, we apply rule (e). Since Chile has the best goal difference of +2 it finishes first among the three teams and so it advances to the second round. b) Justify why Turkey advances to the second round. [5 marks] Let us start our reasoning from the end of our answer to question a). Rule (e) separates Chile from Turkey and Morocco, but does not separate Turkey from Morocco since both of their goal differences among the teams concerned is -1. Therefore, the teams concerned remain Chile, Turkey, and Morocco, and we need to continue breaking this three-way tie. We proceed to rule (f). Since Turkey has scored 3 goals among the teams concerned, and Morocco has scored only 2 goals among the teams concerned, then Turkey finishes second among the three teams and so it advances to the second round. c) Justify why Morocco advances to the second round. [5 marks] Again, let us start our reasoning from the end of our answer to question a). Rule (e) separates Chile from Turkey and Morocco, but does not separate Turkey from Morocco since both of their goal differences among the teams concerned is -1. Therefore, the teams concerned become Turkey and Morocco and we need to break this two-way tie. According to the regulations we need to look at the games played between those two teams to break the tie. The relevant results appear below. Flag Country
Pts W
D L
GF GA
GD
Morocco
3
1
0
0
2
0
+2
Turkey
0
0
0
1
0
2
-2
Morocco 2 Turkey 0 1
2
SOLUTIONS 1
Option 1 : To break the two-way tie we proceed to the next rule, which rule (f). Since Morocco has scored 2 goals among the teams concerned, and Turkey has scored 0 goals among the teams concerned, then Morocco finishes ahead of Turkey. Therefore, Morocco finishes second among the three teams and so it advances to the second round. Option 2 : To break the two-way tie we proceed to the rule that has the highest precedence, which rule (d). Since Morocco has 3 points among the teams concerned, and Turkey has 0 points among the teams concerned, then Morocco finishes ahead of Turkey. Therefore, Morocco finishes second among the three teams and so it advances to the second round. d) Augment the FIFA regulations to remove the ambiguousness of this situation while remaining equally fair to all teams. What two teams advance to the second round according to your regulations? [5 marks] Add the following sentence to the wording that appears between rule (c) and rule (d). “When applying rules (d) to (f), if any of the teams are separated from any of the other teams then break the remaining ties by proceeding to rule (d) and applying the rules separately to each group of teams that were not already separated.” Using these augmented regulations, Morocco is the team that advances to the second round. 2. Since ↓ can be used to define the other operators we only need to define ↓ in terms of ⊕ and →. Using this approach it is easiest to first define ¬ in terms of ⊕ and →. i) ¬p ≡ p → (p ⊕ p) ii) p ↓ q ≡ ¬p → q (And now we can get the remaining operators for “free” by copying from the statement of the question) iii) p ∨ q ≡ ¬(p ↓ q) iv) p ∧ q ≡ ¬p ↓ ¬p v) p ↑ q ≡ ¬(p ∧ q) vi) p ↔ q ≡ (p → q) ∧ (q → p) 3. a) There is more than one possible interpretation for the literal meaning. For example, the sign can be interpreted as stating actions that are permitted or actions that are prohibited. In this solution we will take the literal meaning to be ¬p ∧ ¬q ∧ ¬r. There is more than one possible interpretation for the intended meaning. for example, the meaning can be interpreted to be an implication or an equivalence. In this solution we will take the intended meaning to be p ∧ q ↔ r. p q r ¬p ∧ ¬q ∧ ¬r p ∧ q ↔ r T T T F T T T F F F T F T F F b) T F F F T F T T F F F T F F T F F T F F F F F T T c) By operator precedence (¬, ∧, ↑, ∨, ↓, ⊕, ←, ↔) and associativity (right to left) ¬p ∧ ¬q ∧ ¬r is equivalent to (¬p) ∧ ((¬q) ∧ (¬r)), and (p ∧ q) ↔ r is equivalent to (p ∧ q) ↔ r. Therefore, the formation diagrams appear below:
r
p
p
q
r
q
SOLUTIONS 1
3
d) To obtain the Polish notation we perform a preorder traversal of the formation trees that gives ∧ ¬ p ∧ ¬ q ¬ r (literal) and ↔ ∧ p q r (intended). To obtain reverse Polish notation we reverse these strings to obtain r ¬ q ¬ ∧ p ¬ ∧ (literal) and r q p ∧ ↔ (intended). p q r (p → q) → r p → (q → r) T T T T T T T F F F T F T T T 4. T F F T T F T T T T F T F F T F F T T T F F F F T a) The assignment v where v(p) = F , v(q) = F , v(r) = F provides the result since it gives the interpretations v((p → q) → r) = F and v(p → (q → r)) = T . b) From the above truth table we see that whenever v((p → q) → r) = T then v(p → (q → r)) = T , and so therefore (p → q) → r |= p → (q → r) is true. On the other hand, the assignment in part a) gives an example where v(p → (q → r)) = T and v((p → q) → r) = F , and so p → (q → r) |= (p → q) → r is not true. c) The other operators that are associate are: ∧, ∨, ⊕, ↔. 5. Let’s rewrite A = (p ∨ q) ∧ (r ⊕ (q → ¬p)) as A = (p ∨ q) ∧ (r ⊕ (q → (¬p))) so that the formula’s primary operators will even more obvious in each subformula. Likewise, ¬A = ¬((p ∨ q) ∧ (r ⊕ (q → (¬p)))). (p ∨ q) ∧ (r ⊕ (q → (¬p)))
p ∨ q, r ⊕ (q → (¬p))
p ∨ q, ¬(r → (q → (¬p)))
p ∨ q, ¬((q → (¬p)) → r)
p ∨ q, r, ¬(q → (¬p))
p ∨ q, q → (¬p), ¬r
p ∨ q, ¬q, ¬r
p ∨ q, r, q, ¬(¬p)
p ∨ q, r, q, p
p, r, q, p a)
q, r, q, p
p, ¬q, ¬r
q, ¬q, ¬r
p ∨ q, ¬p, ¬r
p, ¬p, ¬r
q, ¬p, ¬r
4
SOLUTIONS 1
¬((p ∨ q) ∧ (r ⊕ (q → (¬p))))
p
¬(p ∨ q)
¬(r ⊕ (q → (¬p)))
¬p, ¬q
r → (q → (¬p)), (q → (¬p)) → r
¬r, (q → (¬p)) → r
¬r, ¬(q → (¬p)) ¬r, r
q → (¬p), (q → (¬p)) → r
q → (¬p), r
q → (¬p), ¬(q → (¬p))
¬r, q, ¬(¬p)
q → (¬p), q, ¬(¬p))
¬r, q, p
q → (¬p), q, p
¬q, q, p
¬q, r
¬p, r
¬p, q, p
b) c) The semantic tableau for A gives the following information: A T
satisfiable unfalsifiable falsifiable unsatisfiable F
¬A F T
And then the semantic tableau for ¬A fills in the remaining information: A satisfiable T unfalsifiable F falsifiable T unsatisfiable F
¬A T F T F
6. CONSEQ(U ,¬A) is true iff U |= ¬A. In particular, CONSEQ(∅,¬A) is true iff |= ¬A. Likewise, NOTSAT(A) is true iff |= ¬A. Therefore, CONSEQ(∅,¬A) is true iff NOT-SAT(A) is true. Alternate answer: Let’s show that CONSEQ(T rue, ¬A) is true iff NOT-SAT(A). For the first direction assume that CONSEQ(T rue, ¬A) is true. Then for every assignment v in which v(T rue) = T rue then we also have that v(¬A) is true. Since v(T rue) = T rue for all v, this means that v(¬A) is always true. Thus, NOT-SAT(A) is true. Therefore, we have shown that if CONSEQ(T rue, ¬A) is true then NOT-SAT(A) is true. For the other direction, assume that NOT-SAT(A) is true. Then v(A) = F alse for all v. Thus, v(¬A) = T rue for all v. Therefore, CONSEQ(U ,¬A) is true for any U . In particular, CONSEQ(T rue,¬A) is true. Therefore, we have shown that if NOT-SAT(A) is true then CONSEQ(T rue, ¬A) is true. (Note: The same reasoning shows that CONSEQ(A, F alse) is true iff NOT-SAT(A) is true.)
SOLUTIONS 1
7.
5
• Contradiction. Therefore, astronauts do not exist. AstroNOT! • This can be the case as astronauts face in one wealthy direction the entire time before blasting off to space, thereby accruing vast sums of money regardless of absence of direction in space. • When astronauts get paid they are on Earth!
