Discrete Mathematics/Logic/Answers

Answers to Logic Exercise 1
1
 * (a) is not a proposition. (It is a command, or imperative.)


 * (b) and


 * (c) are both propositions.


 * (d) is not a proposition; it's a question.


 * (e) strictly speaking is a propositional function, but many people would say it is a proposition.


 * (f) is not a proposition, because the result can be either true or false, it depends on the values of a & b.

2
 * Noting that p is false (1024 bytes is known as 1KB) and q is true, we have:


 * (a)    "1024 bytes is known as 1MB and a computer keyboard is an example of a data input device".  False.


 * (b)    "(Either) 1024 bytes is known as 1MB or a computer keyboard is an example of a data input device". True.


 * The word Either here is optional; it doesn't have - and doesn't need - an equivalent symbol in Logic.


 * (c) "1024 bytes is not known as 1MB". True.

3
 * (a) x &ge; 50


 * (b) x &le; 40


 * (c) 40 < x < 50


 * (d) x < 50 or x > 40. This is true for all values of x.


 * (e) x &ge; 50 (Note that we don't need to say, in addition, that x > 40; this must be true whenever x &ge; 50.)


 * (f) x &ge; 50 and x &le; 40. This can never be true, whatever the value of x.


 * So (d) is a tautology – it's always true; and (f) is always false.

4
 * (a) I don’t like Maths, but I’m going to spend at least 6 hours a week on Maths.


 * (This sounds much more natural than "I don’t like Maths, and I’m going to spend at least 6 hours a week on Maths.")


 * (b) Either I don’t like Maths, or I’m going to spend at least 6 hours a week on Maths.


 * (c) It’s not true that I don’t like Maths. (Or simply: I do like Maths.)


 * (d) Either I don’t like Maths, or I’m not going to spend at least 6 hours a week on Maths.
 * (It's not very easy to get a natural sounding sentence here. It probably helps to include the word "Either", but it's not essential.)


 * (e) It’s not true that either I like Maths or I’m going to spend at least 6 hours a week on Maths. Or, simply: I neither like Maths, nor am I going to spend at least 6 hours a week on Maths.
 * Alternatively, you can write the answer to (f), which is…


 * (f) I don’t like Maths and I’m not going to spend at least 6 hours a week on Maths.

5
 * (a) (ii)


 * (b) (i) and (iii)


 * (c) (iii)

Back to Logic Exercise 1

Answers to Logic Exercise 2
1

(a)

(b)

(c)

(d)

2
 * Both results columns are the same: T, T, T, F.

3
 * Both results are T, F, F, F

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Answers to Logic Exercise 3
1
 * (a) Yes; both results columns give T, T, T, F


 * (b) No; first is F, T, T, F; second is T, F, F, T

2
 * (a) F, T, T, T, T, T, T, T.


 * This is the negation of T, F, F, F, F, F, F, F which is true only when a, b and c are all true. So the expression is equivalent to &not;(a $$\scriptstyle \wedge$$ b $$\scriptstyle \wedge$$ c)


 * (b) T, T, T, T, F, F, F, F, T, T, T, T, F, F, F, F.


 * This is identical to the column for b, so the expression is equivalent to b.

3
 * (a) a &or; b


 * (b) a &or; b


 * (c) q &or; p

4
 * Result is T, T, T, T. So it is always true

5
 * ((p &or; q) $$\scriptstyle \wedge$$ &not;(p $$\scriptstyle \wedge$$ q)) $$\scriptstyle \wedge$$ &not;r

6
 * In each case, the result is F, F, F, F, T, T, F, F

7

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Answers to Logic Exercise 4
1
 * (a) &not;p


 * (b) &not;s $$\scriptstyle \wedge$$ p


 * (c) p &rArr; r


 * (d) r &rArr; (q $$\scriptstyle \wedge$$ p)


 * (e) I shall work for forty hours this week, or I’ll finish my Coursework Assignment.


 * (f) If I shall not finish my Coursework Assignment, then I shan’t pass Maths.

2
 * The table shows only the result column in each case:

So the results are:


 * (a) Yes, always true


 * (b) No


 * (c) Yes


 * (d) Yes


 * (e) No

3
 * The result column in each case is T, F, T, T. So the propositions are all logically equivalent.

Answers to Logic Exercise 5
1 friend(Jimmy)

2 wealthy(Sue) $$\scriptstyle \wedge$$ clever(Sue)

3 wealthy(Jane) $$\scriptstyle \wedge$$ &not;clever(Jane)

4 friend(Mark) $$\scriptstyle \wedge$$ friend(Elaine)

5 friend(Peter) &rArr; &not;boring(Peter)

6 (wealthy(Jimmy) $$\scriptstyle \wedge$$ &not;boring(Jimmy)) &rArr; friend(Jimmy)

Back to Logic Exercise 5

Answers to Logic Exercise 6
1
 * (a) &exist; x, friend(x) $$\scriptstyle \wedge$$ clever(x)


 * (b) &forall;x, clever(x) &rArr; boring(x)


 * (c) &forall;x, friend(x) &rArr; &not;wealthy(x)
 * OR: &not;(&exist; x, friend(x) $$\scriptstyle \wedge$$ wealthy(x))


 * (d) &exist;x, friend(x) $$\scriptstyle \wedge$$ wealthy(x) $$\scriptstyle \wedge$$ clever(x)


 * (e) &forall;x, (clever(x) $$\scriptstyle \wedge$$ friend(x)) &rArr; boring(x)


 * (f) &forall;x, clever(x) &rArr; (boring(x) &or; wealthy(x))

2
 * (a) popstar(x) is "x is a pop-star"


 * overpaid(x) is "x is overpaid"


 * &forall;x, popstar(x) &rArr; overpaid(x)


 * (b) pilot(x) is "x is an RAF pilot"


 * woman(x) is "x is a woman"


 * &exist;x, pilot(x) $$\scriptstyle \wedge$$ woman(x)


 * (c) student(x) is "x is a student"


 * rolls(x) is "x owns a Rolls-Royce"


 * &forall; x, student(x) &rArr; &not;rolls(x)


 * OR: &not;(&exist; x, student(x) $$\scriptstyle \wedge$$ rolls(x))


 * (d) doctor(x) is "x is a doctor"


 * write(x) is "x can write legibly"


 * &exist; x, doctor(x) $$\scriptstyle \wedge$$ &not;write(x)

Back to Logic Exercise 6

Answers to Logic Exercise 7
1
 * (a) Universe of discourse: {people}


 * programmer is "… is a computer programmer"


 * spreadsheets is "… can understand spreadsheets"


 * &exist; x, programmer(x) $$\scriptstyle \wedge$$ &not; spreadsheets(x)


 * (or if universe of discourse is {computer programmers}, &exist; x, &not;spreadsheets(x))


 * (b) Universe of discourse: {people}


 * prisoner is "… is a prisoner"


 * fairTrial is "… deserves a fair trial"


 * &forall; x, prisoner(x) &rArr; fairTrial(x)


 * (or if universe of discourse is {prisoners}, &forall; x, fairTrial(x))


 * (c) Universe of discourse: {people}


 * intelligent is "… is intelligent"


 * palace is "… supports Crystal Palace Football Club"


 * &exist; x, intelligent(x) $$\scriptstyle \wedge$$ palace(x)


 * (d) Universe of discourse: {people}


 * stupid is "… is stupid"


 * curry is "… likes curry"


 * &exist; x, stupid(x) $$\scriptstyle \wedge$$ &not;curry(x)


 * (e) Universe of discourse: {university students}


 * goodLooking is "… is good-looking"


 * intelligent is "… is intelligent"


 * &forall; x, goodLooking(x) &or; intelligent(x)


 * (f) Universe of discourse: {cars}


 * noisy is "… is noisy"


 * dirty is "… is dirty"


 * &not;(&forall; x, noisy(x) $$\scriptstyle \wedge$$ dirty(x))
 * Or: &exist; x, &not;noisy(x) &or; &not;dirty(x)

2
 * (a) Some Boy Scouts cheat at cards.


 * (b) All people with punk hair cheat at cards.


 * (c) No Boy Scouts have punk hair or cheat at cards.


 * (d) Some people who cheat at cards don’t have punk hair.

3
 * (a) Universe of discourse: {cows}


 * eats is "… eats …"


 * &forall; x, eats(x, grass)


 * (b) Universe of discourse: {people}


 * better is "… is better at Maths than …"


 * &exist; x, better(Harry, x)


 * (c) Universe of discourse: {people}


 * likes is " … likes …"


 * &exist; x, likes(x, the Rolling Stones)


 * (d) Universe of discourse: {people}


 * expects is "… expects …"


 * &not;(&exist; x, expects(x, the Spanish Inquisition))
 * Or: &forall; x, &not;expects(x, the Spanish Inquisition)

Back to Logic Exercise 7