Discrete Mathematics/Set theory/Answers

Answers to Set Theory Exercise 1
1
 * (a) Yes; alphanumeric characters are A…Z, a…z and 0…9
 * (b) No; 'tall' is not well-defined
 * (c) Yes; the set is {12.5}
 * (d) Yes; the empty set
 * (e) No; 'good' is not well-defined

2
 * (a) T
 * (b) F
 * (c) T
 * (d) F; A is a subset of U (which we meet in the next section)
 * (e) F; {even numbers} means the set of all the even numbers, not just those between 2 and 10

3
 * (a) {4, 33, &radic;9}
 * (b) {4, -5, 33, &radic;9}
 * (c) {4, 2/3, -2.5, -5, 33, &radic;9}
 * (d) {&radic;2, π}

4
 * (a) F
 * (b) T
 * (c) F
 * (d) T

5 Examples might include:
 * (a) {London, Paris, Rome, …}
 * (b) {1, 3, 5, 7, …}, but not –3 or –1
 * (c) {5, -5}
 * (d) {3, 27, 243, …}

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Answers to Set Theory Exercise 2
1

2
 * They are all equal.

3
 * (a) True (b) False (c) True

4
 * (a) [[Image:VennExercise2Answer4.jpg|left|350px|thumb]]


 * (b) P &sub; Q; R &sub;Q


 * (c) False

5
 * (a) [[Image:VennExercise2Answer5a.jpg|left|350px|thumb]]


 * (b) [[Image:VennExercise2Answer5b.jpg|left|350px|thumb]]


 * (c) [[Image:VennExercise2Answer5c.jpg|left|350px|thumb]]


 * (d) [[Image:VennExercise2Answer5d.jpg|left|350px|thumb]]

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Answers to Set Theory Exercise 3
1




 * (c)
 * C - B = &oslash;

2
 * (a) F
 * (b) F
 * (c) T

3
 * (a) P &sube; Q
 * (b) Q &sube; P

4

5
 * (a) B &prime;
 * (b) A &cap; B &prime;
 * (c) (A &cup; B) &cap; (A &cap; B) &prime; or (A &cap; B &prime;) &cup; (A &prime; &cap; B)
 * (d) (A &cap; B) &cup; (A &prime; &cap; B &prime;) or (A &cup; B) &prime; &cup; (A &cap; B) or …?

6
 * (a) Region (b) represents A – B. So A – B = A &cap; B &prime;


 * (b) Region (c) represents A &Delta; B.
 * So A &Delta; B = (A &cap; B) &cup; (A &prime; &cap; B &prime;) or (A &cup; B) &prime; &cup; (A &cap; B)

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Answers to Set Theory Exercise 4
1


 * (a) P(A) = { &oslash;, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2,4}, {3, 4}, {2, 3, 4}, {1, 3, 4}, {1, 2, 4}, {1, 2, 3}, {1, 2, 3, 4}}


 * | P(A) | = 16


 * (b) 32


 * (c) 210 = 1024

2

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Answers to Set Theory Exercise 5
1
 * (a) X &times; Y = {(a, a), (a, b), (a, e), (a, f), (c, a), (c, b), (c, e), (c, f)}


 * (b) Y &times; X = {(a, a), (a, c), (b, a), (b, c), (e, a), (e, c), (f, a), (f, c)}


 * (c) X &times; X = {(a, a), (a, c), (c, a), (c, c)}


 * (d) They are equal: A = B

2
 * (a) (b, 2), (b, 4), (c, 1), (c, 5), (e, 1), (e, 5), (f, 2), (f, 4)


 * (b) P = C &times; R


 * (c) ((G &times; R) &cup; (C &times; T)) - (G &times; T)

3
 * V = {pqr | (p, q, r) &isin; L &times; (L &cup; D) &times; (L &cup; D)}

4


 * The shaded area is the same in each case, so it looks as though the proposition is true.

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