BEST NOTES ON CALCULUS (LECTURE5)

The Set N of Natural Numbers

Answer: We denote the set {1, 2, 3,..} of all positive integers by N. Each positive integer n has a successor, namely n + 1. Thus the successor of 2 is 3, and 37 is the successor of 36. You will probably agree that the following properties of N are obvious; at least the first four are. N1. 1 belongs to N. N2. If n belongs to N, then its successor n+ 1 belongs to N. N3. 1 is not the successor of any element in N.

The Set Q of Rational Numbers

Answer: Small children first learn to add and to multiply positive integers. After subtraction is introduced, the need to expand the number sys- tem to include 0 and negative integers becomes apparent. At this point the world of numbers is enlarged to include the set Z of all integers. Thus we have Z {0, 1, -1,2, -2,...}.

The Set R of Real Numbers

Answer: The set Q is probably the largest system of numbers with which you really feel comfortable. There are some subtleties but you have learned to cope with them. For example, Q is not simply the set of symbols m/n, where m, n ∈ Z, n ̸= 0, since we regard some pairs of different looking fractions as equal. For example, 24 and 36 represent the same element of Q. A rigorous development of Q based on Z, which in turn is based on N, would require us to introduce the notion of equivalence classes. In this book we assume a familiarity with and understanding of Q as an algebraic system. However, in order to clarify exactly what we need to know about Q, we set down some of its basic axioms and properties.

Properties of Real Numbers

Answer: A1. a+(b+c)=(a+b)+cforalla,b,c. A2. a+b=b+a for all a,b. A3. a+0=aforalla. A4. For each a, there is an element −a such that a + (−a) = 0. M1. a(bc) = (ab)c for all a, b, c. M2. ab = ba for all a, b. M3. a·1=aforalla. M4. For each a ̸= 0, there is an element a−1 such that aa−1 = 1. DL. a(b+c)=ab+ac fo rall a,b,c.

The Completeness Axiom

Answer: Let S be a nonempty subset of R. (a) If S contains a largest element s0 [that is, s0 belongs to S and s≤s0 foralls∈S],thenwecalls0 themaximumofSand write s0 = maxS. (b) If S contains a smallest element, then we call the smallest element the minimum of S and write it as min S.