Exercise 1.2 answers: Sets

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(i) odd natural numbers divisible by 2

1. An odd number can never be divisible by 2, so no number satisfies both conditions.

AnswerYes, it is the null set.

(ii) even prime numbers

1. 2 is a prime number and it is also even.
2. So the set holds 2 and is not empty.

AnswerNo, it is not the null set; it is {2}.

(iii) x < 5 and x > 7

1. No number can be both less than 5 and greater than 7 at the same time.

AnswerYes, it is the null set.

(iv) a point common to any two parallel lines

1. Two parallel lines never meet, so they share no point.

AnswerYes, it is the null set.

Watch this explained “The sort, and two look-alikes”, 11:01 into A set with nothing in it, and sets you cannot finish listing

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(i) months of a year

1. A year has exactly 12 months.

AnswerFinite

(ii) {1, 2, 3, . . .}

1. This list keeps going forever with no last natural number.

AnswerInfinite

(iii) {1, 2, 3, . . .99, 100}

1. This set stops at 100, giving exactly 100 members.

AnswerFinite

(iv) positive integers greater than 100

1. There is no largest integer, so this set never stops.

AnswerInfinite

(v) prime numbers less than 99

1. Only finitely many primes lie below 99 − they can all be listed.

AnswerFinite

Watch this explained “The sort, and two look-alikes”, 11:01 into A set with nothing in it, and sets you cannot finish listing

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(i) lines which are parallel to the x-axis

1. A line parallel to the x-axis can be drawn at any height, and there are endlessly many heights.

AnswerInfinite

(ii) letters in the English alphabet

1. The English alphabet has exactly 26 letters.

AnswerFinite

(iii) numbers which are multiple of 5

1. Multiples of 5 keep going forever: 5, 10, 15, and so on.

AnswerInfinite

(iv) animals living on the earth

1. Nobody has counted every animal alive right now, but a definite number of them exists at any moment.
2. Not being able to count something is not the same as it never ending.

AnswerFinite

(v) circles passing through the origin (0,0)

1. A circle through the origin can have any centre and radius as long as it passes through (0,0) − endlessly many choices.

AnswerInfinite

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(i) A = { a, b, c, d }

1. Both sets hold exactly a, b, c, d − only the order written is different, and order does not matter for a set.

AnswerA = B

(ii) A = { 4, 8, 12, 16 }

1. A has 12 but B does not; B has 18 but A does not.

AnswerA ≠ B

(iii) x is positive even integer and x ≤ 10

1. The positive even integers up to 10 are 2, 4, 6, 8, 10 − the same as A.

AnswerA = B

(iv) x is a multiple of 10

1. B contains 15 and 25, and neither is a multiple of 10.

AnswerA ≠ B

Watch this explained “The check, and it has two halves”, 0:51 into Why order and repetition cannot make two sets different

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(i) x is solution of x² + 5x + 6 = 0

1. Factorise: x² + 5x + 6 = (x + 2)(x + 3).
2. So B = {−2, −3}, but A = {2, 3} − the signs are opposite.

AnswerNot equal; A = {2, 3} but B = {−2, −3}.

(ii) x is a letter in the word FOLLOW

1. FOLLOW gives the distinct letters F, O, L, W.
2. WOLF gives the same four letters F, O, L, W.

AnswerEqual; both sets are {F, O, L, W}.

Watch this explained “The sign trap”, 9:16 into Why order and repetition cannot make two sets different

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1. B = {1,2,3,4} and D = {3,1,4,2} hold exactly the same four numbers, only written in a different order.
2. E = {−1,1} and G = {1,−1} likewise hold exactly the same two numbers.
3. A and C share 4, 8, 12, but A also has 2 (not in C) and C also has 14 (not in A), so they are not equal.
4. F holds an unnamed element a; with nothing said about what a is, F cannot be matched to H.

AnswerB = D and E = G; no other pair among these eight sets is equal.

Watch this explained “Eight sets, sorted into families”, 10:18 into Why order and repetition cannot make two sets different