Sets N, R, C, Z, and QDate: 01/22/2001 at 06:40:57 From: Fenja, Eva & Liliane Subject: Sets N, R, C, Z, Q Hello Dr. Math, Could you help us with these questions? 1) What are the exact and extensive definitions of the sets N, R, C, Z and Q? 2) What proportion do these sets bear to one another? We thank you in advance. Yours sincerely, Fenja, Eva and Liliane (from Holland) Date: 01/22/2001 at 14:09:46 From: Doctor Rob Subject: Re: Sets N, R, C, Z, Q Thanks for writing to Ask Dr. Math, Fenja, Eva, and Liliane. 1) The smallest set N that satisfies the following postulates is indistinguishable from, and can be taken to be, the natural numbers: P1. 1 is in N. P2. If x is in N, then its "successor" x' is in N. P3. There is no x in N such that x' = 1. P4. If y in N isn't 1, then there is a x in N such that x' = y. P5. If x and y are in N and x' = y', then x = y. P6. If S is a subset of N, 1 is in S, and the implication (x in S ==> x' in S) holds, then S = N. This is the recursive definition of addition: D+: Let a and b be in N. (1) If b = 1, then define a + b = a' (using P1 and P2). (2) If b isn't 1, then let c' = b, with c in N (using P4), and define a + b = (a + c)'. This can be restated as (1) a + 1 = a', and (2) a + b' = (a + b)'. This is the recursive definition of multiplication: D*: Let a and b be in N. (1) If b = 1, then define a * b = a. (2) If b isn't 1, then let c' = b, with c in N (using P4), and define a * b = (a * c) + a. This can be restated as (1) a * 1 = a, and (2) a * b' = (a * b) + a. This is the definition of the order: D<: Let a and b be in N. a < b if and only if there exist c in N such that b = a + c. It is a long exercise to prove that the natural numbers obey the commutate, associative, and distributive laws, that 1 is the multiplicative identity, and that order is respected by addition and multiplication. Recall that a relation R on a set S is a subset of the Cartesian product S x S, that is, R is a subset of the set of all pairs of elements of S. Recall that an equivalence relation is a relation which is reflexive, symmetric, and transitive, that is, satisfies the following three properties: (1) For all a, (a,a) is in R. (2) For all a and b, (a,b) in R implies that (b,a) is in R. (3) For all a, b, and c, (a,b) and (b,c) both in R implies that (a,c) is in R. For an equivalence relation R we often denote (x,y) in R by x ~ y. Then the set Z of integers is defined to be the set of all equivalence classes of pairs of natural numbers (a,b), where the equivalence relation is (a,b) ~ (c,d) if and only if a + d = b + c. The class containing (1,1) is called "zero" and denoted "0". The class containing (a',1) can be identified with the natural number a in N, and is so denoted. The class containing (1,a') is called "negative a" and is denoted by "-a". One defines addition, multiplication, and the ordering in Z by (a,b) + (c,d) = (a+c, b+d), (a,b) * (c,d) = (a*c+b*d, a*d+b*c), (a,b) < (c,d) if and only if a + d < b + c. One defines subtraction by a - b = a + (-b). It is again an exercise to prove that the integers obey the commutative, associative, and distributive laws, 1 is the multiplicative identity, 0 is the additive identity, -a is the additive inverse of a, and that the order is respected by addition and multiplication. Furthermore the pair (a,b) is an element of the equivalence class a - b. Then the set Q of rational numbers is defined to be the set of all equivalence classes of pairs of integers (a,b) with b not zero, where the equivalence relation is (a,b) ~ (c,d) if and only if a*d = b*c. The class containing (a,1) can be identified with the integer a in Z, and is so denoted. One defines addition, multiplication, and the ordering in Q by (a,b) + (c,d) = (a*d+b*c, b*d), (a,b) * (c,d) = (a*c, b*d), (a,b) < (c,d) if and only if 0 < b*d*(b*c-a*d). It is again an exercise to prove that the rational numbers obey the commutative, associative, and distributive laws, 1 is the multiplicative identity, 0 is the additive identity, -a is the additive inverse of a, if a isn't zero, then 1/a is the multiplicative inverse of a, and the ordering is respected by addition and multiplication. One defines subtraction by a - b = a + (-b). One defines division by a/b = a*(1/b). Then the pair (a,b) is an element of the equivalence class a/b. Then the set R of real numbers is defined to be the set of all equivalence classes of pairs of subsets of Q (A,B) such that Q = A U B, and for every a in A and every b in B, a < b. Such pairs are called "Dedekind cuts." The equaivalence relation is (A,B) ~ (C,D) if and only if #(A-C) <= 1 and #(B-D) <= 1. The rational number a can be identified with the equivalence class containing the pair (A,B) where A = {x in Q: x < a} and B = Q - A. It is again an exercise to prove that the real numbers obey the commutative, associative, and distributive laws, 1 is the multiplicative identity, 0 is the additive identity, -a is the additive inverse of a, if a isn't zero, then 1/a is the multiplicative inverse of a, and the ordering is respected by addition and multiplication. Then the set C of complex numbers is defined to be the set of all pairs or real numbers (a,b), with the following definitions of addition and multiplication: (a,b) + (c,d) = (a+c, b+d), (a,b) * (c,d) = (a*c-b*d, a*d+b*c). The real number a can be identified with the pair (a,0). The pair (0,1) satisfies (0,1)^2 = (-1,0) = -1, and so we denote (0,1) by "i". We write (a,b) = a + b*i. It is again an exercise to prove that the complex numbers obey the commutative, associative, and distributive laws, 1 is the multiplicative identity, 0 is the additive identity, -a is the additive inverse of a, and if a isn't zero, then 1/a is the multiplicative inverse of a. NOTE: There is an alternative definition of the real numbers as the set of equivalence classes of Cauchy sequences of rational numbers. A sequence of rational numbers is a function f from N to Q. A Cauchy sequence or rational numbers is a sequence of rational numbers such that given any e > 0 in Q, there is an x in N such that for all m > x and n > x in N, |f(m)-f(n)| < e. In English, this means that the values of f get arbitrarily close together as its argument gets large. Two Cauchy sequences f and g are called equivalent if, for every e > 0 in Q, there is an x in N such that for all n > x in N, |f(n)-g(n)| < e. In English, this means that the values of f and the values of g get arbitrarily close together as their common argument gets large. Any a in Q can be identified with the equivalence class containing the Cauchy sequence defined by f(n) = a for every n in N. The Dedekind cut (A,B) can be derived from this by letting A be the set of rational numbers which are smaller than all but a finite number of values of the Cauchy sequence f, and B = Q - A. It is also an exercise to show that each of the containments N < Z < Q < R < C is proper, in that there is always an element in the right-hand set which is not in the left-hand one. 2) Let #S be the cardinal number of the set S. Then #C = #R, #R = (uncountable infinity) * #Q, #Q = #Z = #N. The density of N in Z is 1/2. The density of Z in Q is 0. The density of Q in R is 0. The density of R in C is 0. These may not be just what you wanted for the answer to (2), however. If you need more assistance with this, write again. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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