Solutions to Exercises A Solution 1: Symbolic connectives 6 ≠ 3 1. (a) 8 4 12 = 6 = 3 12 = 6 6 = 3 (b) 16 8 4 ,thatis‘16 8 and 8 4 ’. Only (b) is true. 2. r ∧ s 3. p is ‘Two is a prime number’ q is ‘Two is an even number’ p ∨ q Solution 2: Arithmetic operators 1. 3 + 1 + 4 or, more accurately, 3 + (1 + 4).Leftoperand= 3andright operand = 5soresult= 3 + 5 = 8. 2. (2 + 3) − (7 − 5) = 5 − 2 = 3but(7 − 5) − (2 + 3) = 2 − 5 =−3, which is different. Solution 3: Conjunction 1. ‘My cat is black but your cat is white’ ∧ = ∧ = ‘My cat is black’ ‘Your cat is white’ T F F T F 2. ‘Shakespeare wrote both Hamlet and MacBeth’ ∧ = ∧ = ‘Shakespeare wrote Hamlet’ ‘Shakespeare wrote MacBeth’ T T T T T ∧ = ∧ = 3. q p T T F T F 4. 2 × 7 = 27 ∧ 32 = 9 = ∧ = T F T T F ∧ = ∧ = 5. 3 > 2 3 > 2 T T T T T 217 218 A Solutions to Exercises Solution 4: Negation = ¬ = 1. ‘Themoonisnotmadeofbluecheese’ T F T T . = ¬ = 2. ‘1234 is an not even number’ T T T F . Note that ‘1234 is an odd number’ would be incorrect answer to this ques- tion as it depends upon the additional fact that an odd number is one which is not even. Solution 5: Disjunction = ∨ = 1. ‘Either you broke the window or else I’m a Martian’ T F F T F . Note that in general use this is just a more emphatic way of saying ‘you broke the window’. = ∨ = 2. ‘Either Shakespeare or Francis Bacon wrote Hamlet’ T T F T T . ∨ = ∨ = 3. r1 r2 T F T T T = ∨ = 4. ‘Either 2 is even or 3 is odd’ T T T T T Solution 6: Connective schemas 1. (a) ‘Rex does not have a wet nose’ (b) 2 + 3 ≠ 8 (c) ¬p (d) ¬q (e) ¬r2 (f) ‘Rex is black but Rover is white’ (g) ‘Either 2 + 3 = 8 or 32 = 9’ (h) p ∧ q (i) p1 ∧ p2 (j) r2 ∨ q 2. The following are possible answers. Note how some of the meaning of the original may be lost. (a) P∧Qwhere •Pis interpreted as ‘This book is long’; •Qis interpreted as ‘Ireaditquickly’. (b) P∨Qwhere •Pis ‘There is a hole in the exhaust’ •Qis ‘A bracket has worked loose’. (c) P∨Qwhere •Pis ‘Lunch will be served during the flight’. •Qis ‘Dinner will be served during the flight’. (d) ¬P where P is 3 × 6 = 7 A Solutions to Exercises 219 Solution 7: Instantiation to compound propositions 1. ¬¬‘Rover is a brave dog’ 2. 2 ≤ 3 ∧ 22 ≤ 32 ∨ 2 ≥ 3 ∧ 22 ≥ 32 3. p1 ∧¬p2 ∧¬p3 ∧ p4 , though strictly speaking this should be written as (p1 ∧¬p2) ∧ (¬p3 ∧ p4) – see sections 2.9–2.12. Solution 8: Constructing parse trees Main connectives are shown in bold. 220 A Solutions to Exercises 5. ¬((p ∨ q) ∧ r) 6. ((¬p) ∧ (¬q)) ∨∨∨ (¬r) Solution 9: Compound propositions from parse trees 1. q ∧ (p ∧ r) 2. ¬¬(p ∧ q) Solution 10: Connective priorities 1. (a) ¬(¬p) (b) (¬p) ∧ q (c) q ∧ (¬r) (d) (¬p) ∨ q (e) (p ∧ q) ∧ (¬r) (f) (p ∨ q) ∨ (¬r) (g) (q ∧ (¬p)) ∨ q (h) (p1 ∨ p2) ∨ (¬p3) (i) 3 > 0 ∨ ((¬ 1 + 1 = 2) ∧ 2 + 3 = 5) (j) (¬‘Fido has three legs’) ∨ ‘Rex has four legs’ (k) (p ∧ q) ∨ ((¬r)∧ p) (l) (¬p) ∧ ((¬q) ∨ (p ∧ r)) (m) ¬(¬((¬p2) ∧ (¬p1))) A Solutions to Exercises 221 2. The main connective is shown in bold. (a) ¬(p1 ∧ p2) ∨∨∨ p3 (b) ‘Rex has four legs’ ∨ ‘Fido has three legs’ ∨∨∨¬‘Rover has a wet nose’ (c) ¬ ¬q (d) ¬ 1 + 1 = 2 ∨∨∨ (1 + 1)2 = 22 (e) ¬(p ∨ q ∧ r ∨¬s) (f) ¬(¬(¬p1 ∨ p2 ∧ p3) ∨¬p4) ∧∧∧¬(p5 ∧ p6 ∨¬¬p7) Solution 11: Removing parentheses 1. ¬p ∨ q 2. p1 ∨¬p2 ∨ p3 3. p ∧¬q ∨¬r 4. ¬q1 ∧ (q1 ∨ q2) ∨ q1 ∧ q2, though perhaps ¬q1 ∧ (q1 ∨ q2) ∨ (q1 ∧ q2) is easier to read 5. r ∧ (¬p ∨ q) Solution 12: Truth values of compound propositions ¬ ¬ = ¬ = 1. ( T ) T F T T ¬ ∨ = ∨ = 2. T F T F F T F ∨ ∨ = ∨ = 3. (F F ) F T F F T F 4. ¬¬(T ∨¬F ) ∧¬(F ∨ F ) = ¬¬ ∨ ∧¬ T (T T ) F = ¬¬ ∧ T T T = ¬ ∧ T F T = ∧ T T T = T T ¬ ∨ = ∨ = 5. F T T T T T T ¬ ∧ ∨¬ = ∧ ∨ = ∧ = 6. F (T T ) T T (T F ) T T T T T Solution 13: Compound propositions from propositional forms 1. ‘Rex has four legs and either Fido has three legs or Rover does not have a wet nose.’ 2. (a) ‘It is not cold but it is snowing.’ (b) ‘Either the water pump is not working or there is no anti-freeze in the radiator and last night was very cold.’ 222 A Solutions to Exercises Solution 14: Propositional forms for compound propositions Note that the choice of schematic letters is not unique. 1. One possible answer is: ¬P ∧ ¬Q , P∧¬Q, ¬P ∧ Q , P∧Q, P Another possibility is: ¬P1 ∧¬P2 , Q1 ∧¬P2 , ¬P1 ∧Q2 , Q1 ∧Q2 , R In the first answer, occurrences of a schematic letter in one propositional form would not always be instantiated to the same proposition as occur- rences of the same letter in another propositional form in the list: thus the schematic letter P in the propositional form ¬P ∧ ¬Q would need to be instantiated to ‘The sun is made of gold’whereasin P∧¬Q it would need to be instantiated to ‘The sun is not made of gold’. In the second answer, however, all occurrences of a schematic letter would need to be instantiated to the same proposition wherever it occurred in the list: thus P1 would always need to be instantiated to ‘The sun is made of gold’. 2. One possible answer is: (P∧Q) ∨R, P∨R, P Another possibility is: (P1 ∧P2) ∨P3 , Q1 ∨P3 , R1 3. One possible answer is: ¬P ∨ Q , P∨Q, P Another possibility is: ¬P1 ∨P2 , Q1 ∨P2 , R1 4. (P1 ∧P2) ∧ (P3 ∧P4) , Q1 ∧ (P3 ∧P4) , (P1 ∧P2) ∧Q2 , Q1 ∧Q2 , R1 or (P∧Q) ∧ (R∧S) , P∧(R∧S) , (P∧Q) ∧R, P∧R, P 5. P∨¬P P∨Q R or P∨¬P P∨Q P 6. (P1 ∧¬P2)∨(¬P1 ∧P2) , (P1 ∧Q2)∨(¬P1 ∧P2) , (P1 ∧¬P2)∨(Q1 ∧P2) , (P1 ∧Q2) ∨ (Q1 ∧P2) , R1 ∨ (Q1 ∧P2) , (P1 ∧Q2) ∨R2 , R1 ∨R2 , S1 or (P∧¬Q)∨(¬P∧Q)(P∧R)∨(¬P∧Q)(P∧¬Q)∨(S∧Q)(P∧R)∨(S∧Q) P∨(S∧Q)(P∧R) ∨Q P∨Q P Notice care must be taken over the choice of schematic letters, since the characteristic form has repeated occurrences of the same letter. For ex- ample: in the second answer, the propositional form (P∧Q) ∨ (¬P ∧ Q) would not be correct since the letter Q would need to be instantiated to different propositions in that one form – this is not acceptable. Solution 15: Truth tables 1. P ¬P ¬¬P T F T F T F 2. P ¬P P∨¬P T F T F T T A Solutions to Exercises 223 3. Q ¬Q ¬¬Q ¬¬¬Q T FT F F TF T 4. PQ¬Q P ∨ ¬Q ¬(P∨¬Q) TT FT F TF TT F FT FF T FF TT F 5. PQ¬Q P∧¬Q TT F F TF T T FT F F FF T F 6. P1 P2 P3 P1 ∧P2 (P1 ∧P2) ∧P3 TTT T T TTF T F TFT F F TFF F F FTT F F FTF F F FFT F F FFF F F 7. PQRP∧R Q∧(P∧R) TTT T T TTF F F TFT T F TF F F F FTT F F FTF F F FFT F F FFF F F 224 A Solutions to Exercises 8. PQRP∨Q (P∨Q) ∨R TTT T T TT F T T TFT T T TFF T T FTT T T FTF T T FFT F T FFF F F 9. PQRQ∨R P∨(Q∨R) TTT T T TT F T T TFT T T TFF F T FTT T T FTF T T FFT T T FFF F F 10. PQRP∨Q (P∨Q) ∧R TTT T T TT F T F TFT T T TFF T F FTT T T FTF T F FFT F F FFF F F 11. PQRP∧R Q∧R (P∧R) ∨ (Q∧R)) TTT TT T TT F FF F TFT TF T TFF FF F FTT FT T FTF FF F FFT FF F FFF FF F A Solutions to Exercises 225 12. Q1 Q2 Q3 Q2 ∧Q3 Q1 ∨ (Q2 ∧Q3) TTT T T TTF F T TFT F T TFF F T FTT T T FTF F F FFT F F FFF F F 13. PQRP∨Q P∨Q (P∨Q) ∧ (P∨R) TTT TT T TTF TT T TFT TT T TF F TT T FTT TT T FTF TF F FFT FT F FFF FF F 14. PQRQ∧R P∨(Q∧R) ¬(P∨(Q∧R)) TTT TT F TTF FT F TFT FT F TF F FT F FTT TT F FTF FF T FFT FF T FFF FF T 15. PQR¬Q ¬R ¬Q ∨ ¬R ¬P ¬P ∧ (¬Q ∨ ¬R) TTT FF F F F TTF FT T F F TFT TF T F F TF F TT T F F FTT FF F T F FTF FT T T T FFT TF T T T FFF TT T T T 226 A Solutions to Exercises 16. PQRQ∨R P∧(Q∨R) ¬(P∧(Q∨R)) TTT TT F TT F TT F TFT TT F TFF FF T FTT TF T FTF TF T FFT TF T FFF FF T 17.