For one reason, many of these ideas will be already familiar to the students — at least informally. If the students have already studied abstract algebra, number theory or com- binatorics, they should be familiar with the use of mathematical induction. If not, then some time should be spent on mathematical induction. However, we believe that it is not necessary to go into the proofs of these results at this time. Section 1.

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For one reason, many of these ideas will be already familiar to the students — at least informally. If the students have already studied abstract algebra, number theory or com- binatorics, they should be familiar with the use of mathematical induction. If not, then some time should be spent on mathematical induction. However, we believe that it is not necessary to go into the proofs of these results at this time.

Section 1. This type of element-wise argument is very common in real analysis, since manipulations with set identities is often not suitable when the sets are complicated. Sample Assignment: Exercises 1, 3, 9, 14, 15, Partial Solutions: 1. The opposite inclusion is shown in Example 1. The proof for unions is similar. Show that this function works.

Many examples are possible. See Example 1. Thus f is injective. Thus g is surjective. Since may students have only a hazy idea of what is involved, it may be a good idea to spend some time explaining and illustrating what constitutes a proof by induction.

The truth of falsity of the individual assertion is not an issue here. Sample Assignment: Exercises 1, 2, 6, 11, 13, 14, Probably every instructor will want to show that Q is countable and R is uncountable see Section 2. The teacher must avoid getting bogged down in a protracted discussion of cardinal numbers. Sample Assignment: Exercises 1, 5, 7, 9, If b also maps into 1, then c must map into 2; if b maps into 2, then c can map into either 1 or 2.

Thus there are 3 surjections that map a into 1, and there are 3 other surjections that map a into 2. The bijection of Example 1. There are also 2n subsets of type ii. See Exercise Therefore Theorem 1. Related Papers.

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## INTRODUCTION TO REAL ANALYSIS ROBERT G BARTLE SOLUTIONS PDF

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