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- Question : EX3.1 - The field axioms include rules known often as associative rules, commutative rules and distributive rules.Which are which and why do they have these names?
- Question : EX3.2 - To be precise we would have to say what is meant by the operations of addition and multiplication.Let S be a set and let S
- Question : EX3.3 - 3 If in the field axioms for R we replace R by any other set with two operations + and
- Question : EX3.4 - Let S be a set consisting of two elements labeled as A and B.Define A+A = A, B + B = A, A + B = B + A = B, A
- Question : EX3.5 - Using just the field axioms, show that (x + 1)2 = x2 + 2x + 1 for all x ? R.Would this identity be true in any field?
- Question : EX3.6 - Define operations of addition and multiplication on Z5 = {0, 1, 2, 3, 4} as follows:
- Question : EX3.7 - 7 Define operations of addition and multiplication on Z6 = {0, 1, 2, 3, 4, 5} as follows:
- Question : EX4.1 - 7 Define operations of addition and multiplication on Z6 = {0, 1, 2, 3, 4, 5} as follows:
- Question : EX4.2 - Show for every n ? IN that n2 ? n.
- Question : EX4.3 - Using just the axioms, prove the arithmetic-geometric mean inequality: ? ab ? a + b 2 for any a, b ? R with a > 0 and b > 0.(Assume, for the moment, the existence of square roots.)
- Question : EX6.1 - Show that a set of real numbers E is bounded if and only if there is a positive number r so that |x| < r for all x ? E.
- Question : EX6.2 - Find sup E and inf E and (where possible) max E and min E for the following examples of sets: (a) E = IN (b) E = Z
- Question : EX6.3 - Under what conditions does sup E = max E?
- Question : EX6.4 - Show for every nonempty, finite set E that sup E = max E.
- Question : EX6.5 - For every x ? R define [x] = max{n ? Z : n ? x} called the greatest integer function. Show that this is well defined and sketch the graph of the function.
- Question : EX6.6 - Let A be a set of real numbers and let B = {?x : x ? A}.Find
- Question : EX6.7 - Let A be a set of real numbers and let B = {?x : x ? A}.Find a relation between sup A and inf B and between inf A and sup B
- Question : EX6.8 - Let A be a set of real numbers and let B = {x+r : x ? A} for some number r.Find a relation between sup A and sup B.
- Question : EX6.9 - 9 Let A be a set of real numbers and let B = {xr : x ? A} for some positive number r.Find a relation between sup A and sup B.(What happens if r is negative?)
- Question : EX6.10 - Let A and B be sets of real numbers such that A ? B.Find a relation among inf A, inf B, sup A, and sup B.
- Question : EX6.11 - 1 Let A and B be sets of real numbers and write C = A ? B.Find a relation among sup A, sup B, and sup C.
- Question : EX6.12 - Let A and B be sets of real numbers and write C = A ? B.Find a relation among sup A, sup B, and sup C.
- Question : EX6.13 - Let A and B be sets of real numbers and write C = {x + y : x ? A, y ? B}. Find a relation among sup A, sup B, and sup C.
- Question : EX6.14 - Let A and B be sets of real numbers and write C = {x + y : x ? A, y ? B}. Find a relation among inf A, inf B, and inf C.
- Question : EX6.15 - Let A be a set of real numbers and write A2 = {x2 : x ? A}.Are there any relations you can find between the infs and sups of the two sets?
- Question : EX6.16 - Let E be a set of real numbers.Show that x is not an upper bound of E if and only if there exists a number e ? E such that e>x.
- Question : EX6.17 - 7 Let A be a set of real numbers.Show that a real number x is the supremum of A if and only if a ? x for all a ? A and for every positive number ? there is an element a ? A such that x ? ?
- Question : EX6.18 - Formulate a condition analogous to the preceding exercise for an infimum.
- Question : EX6.19 - Using the completeness axiom, show that every nonempty set E of real numbers that is bounded below has a greatest lower bound (i.e., inf E exists and is a real number).
- Question : EX6.20 - 0 A function is said to be bounded if its range is a bounded set.Give examples of functions f : R ? R that are bounded and examples of such functions that are unbounded.Give an example of one that has the property that sup{f(x) : x ? R} is finite but max{f(x) : x ? R} does not exist.
- Question : EX6.21 - The rational numbers Q satisfy the axioms for an ordered field.Show that the completeness axiom would not be satisfied.That is show that this statement is false: Every nonempty set E of rational numbers that is bounded above has a least upper bound (i.e., sup E exists and is a rational number)
- Question : EX6.22 - Let F be the set of all numbers of the form x + ?2y, where x and y are rational numbers.Show that F has all the properties of an ordered field but does not have the completeness property.
- Question : EX6.23 - Let A and B be nonempty sets of real numbers and let ?(A, B) = inf{|a ? b| : a ? A, b ? B}. ?(A, B) is often called the
- Question : EX7.1 - Using the archimedean theorem, prove each of the three statements that follow the proof of the archimedean theorem
- Question : EX7.2 - Suppose that it is true that for each x > 0 there is an n ? IN so that 1/n < x. Prove the the archimedean theorem using this assumption.
- Question : EX7.3 - Without using the archimedean theorem, show that for each x > 0 there is an n ? IN so that 1/n < x.
- Question : EX7.4 - Let x be any real number.Show that there is an integer m ? Z so that m ? x
- Question : EX7.5 - 5 The mathematician Leibniz based his calculus on the assumption that there were
- Question : EX7.6 - The archimedean property asserts that if x > 0, then there is an integer N so that 1/N < x.The proof requires the completeness axiom.Give a proof that does not use the completeness axiom that works for x rational.Find a proof that is valid for x = ?y, where y is rational.
- Question : EX7.7 - In Section 1.2 we made much of the fact that there is a number whose square is 2 and so ?2 does exist as a real number.Show that ? = sup{x ? R : x2 < 2} exists as a real number and that ?2 = 2
- Question : EX8.1 - Show that any bounded, nonempty set of natural numbers has a maximal element.
- Question : EX8.2 - Show that any bounded, nonempty subset of Z has a maximum and a minimum
- Question : EX8.3 - For further exercises on proving statements using induction as a method, see Section A.8.
- Question : EX9.1 - Show that the definition of
- Question : EX9.2 - Find a rational number between ?10 and ?.
- Question : EX9.3 - If a set E is dense, what can you conclude about a set A ? E?
- Question : EX9.4 - 4 If a set E is dense, what can you conclude about the set R \ E?
- Question : EX9.5 - If two sets E1 and E2 are dense, what can you conclude about the set E1?E2?
- Question : EX9.6 - 6 Show that the dyadic rationals (i.e., rational numbers of the form m/2n for m ? Z, n ? IN) are dense.
- Question : EX9.7 - 7 Are the numbers of the form
- Question : EX9.8 - 8 Show that the numbers of the form
- Question : EX10.1 - Show that |x| = max{x, ?x}.
- Question : EX10.2 - Show that max{x, y} = |x ? y|/2+(x + y)/2.What expression would give min{x, y}
- Question : EX10.3 - Show that the inequalities |x ? a| < ? and a ? ?
- Question : EX10.4 - Show that if ?
- Question : EX10.5 - Show that ||x|?|y|| ? |x ? y| assuming the triangle inequality (i.e., that |a + b|?|a| + |b|).This inequality is also called the triangle inequality.
- Question : EX10.6 - Under what conditions is it true that |x + y| = |x| + |y|?
- Question : EX10.7 - Under what conditions is it true that |x ? y| + |y ? z| = |x ? z|?
- Question : EX10.8 - Show that |x1 + x2 +
- Question : EX10.9 - Let E be a a set of real numbers and let A = {|x| : x ? E}.What relations can you find between the infs and sups of the two sets?
- Question : EX10.10 - Find the inf and sup of the set {x : |2x + ?| < ?2
- Question : EX10.11 - The complex numbers C are defined as equal to the set of all ordered pairs of real numbers subject to these operations: (a1, b1)+(a2, b2)=(a1 + a2, b1 + b2) and (a1, b1)
- Question : EX10.12 - Can an order be defined on the field C of Exercise 1.11.1 in such a way so to make it an ordered field?
- Question : EX10.13 - The statement that every complete ordered field
- Question : EX10.14 - 4 We have assumed in the text that the set IN is obviously contained in R. After all, 1 is a real number (it
- Question : EX10.15 - Use this definition of
- Question : EX10.16 - Let G be a subgroup of the real numbers under addition (i.e., if x and y are in G, then x + y ? G and ?x ? G).Show that either G is a dense subset of R or else there is a real number ? so that G = {n? : n = 1,

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