By John Baylis, Rod Haggarty

ISBN-10: 0333442423

ISBN-13: 9780333442425

ISBN-10: 134909532X

ISBN-13: 9781349095322

'...quite the easiest one i've got had the fortune to read...admirable substitute interpreting for a beginning direction introducing collage mathematics.' David Tall, the days greater academic complement

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**Additional info for Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics**

**Example text**

We aim to scatter throughout this book properties of IR which can only be described as mind-boggling, and no amount of careful thinking about lines would have revealed them. Even hindsight is not enough. You may decide, for example, that the trouble with open intervals is that they are just that little bit smalIer than the corresponding closed intervals, and since we are talking about sequences of intervals whose lengths are approaching zero, 48 THE REAL NUMBERS that little bit may be enough to tip the balance .

In particular (and appealing to our first digression), it follows that all the pairs have the same longest common measure. By the way the successive segments were constructed we have: and CD ~ 2CD I, COl ~ 2CD 2, CO 2 ~ 2CD 3 , · · · and from these strings of inequalities it is clear that AB ~ 2A IB ~ 4A 2B ~ 8A 3B ~ .. and CD ~ 2CD I ~ 4CD 2 ~ 8CD 3 ~ ••• so and CD CD;_I::;; 2;-1 Hence, A;B and CD;-1 can be made as small as we like by taking i sufficiently big-that is, taking the construction through sufficiently many steps.

30] Do the same for (a, b)fYi(e,d) if and only if ad = be, where fYi is a relation on 7L. x (7L. \ {0 }). 3 should recognise the equivalence classes involved in the two examples above! 31] How many different equivalence relations are there on a threeelement set? 32] A relation fYi on {a, b, c, d} consists of the ordered pairs (a, b), (a, c), (a, a), (b, d), (e, c). 0// (i) reflexive, (ii) symmetric, (iii) transitive. Then do the same for the set { (a, b), (a, c), (a, a), (e, c) }. 33] Let n be a fixed natural number and IX be a relation defined on ~ by aab if and only if a 2 - b 2 is a multiple of n.

### Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics by John Baylis, Rod Haggarty

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