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- Use the axioms for a field to prove that 0 .
*a*= 0 for any element*a*of a field.Prove that -1 . -1 = 1 and thereby justify the old rhyme:

*Minus times minus is equal to plus*

The reasons for this we will not discuss.Use the axioms for an ordered field to prove that 1 > 0 in any ordered field.

Suppose that

*a*,*b*,*c*are elements of an ordered field. Prove the following.

if*a*> 0 then 0 > -*a*,

if*a*> 0 and 0 >*b*then 0 >*a*.*b*,

if*a*>*b*and 0 >*c*then*b*.*c*>*a*.*c*,

if*a*> 0 then*a*^{-1}> 0, if 0 >*a*then 0 >*a*^{-1}. - Let
*F*be the set {0, 1, 2, 3, 4, 5, 6} on which addition and multiplication are defined modulo 7. Show that*F*is a field under these operations.Define an ordering 6 > 5 > 4 > 3 > 2 > 1 > 0 on

*F*and show that*F*is not an ordered field under this ordering.

Prove that*F*is not an ordered field under any ordering. - Write down the least upper bound (lub) and greatest lower bound (glb) of each of the following sets.

In which cases are the lub and glb elements of the set?(a) {

*x*∈**Q**|*x*^{3}< 2}

(b) {*x*∈**R - Q**|*x*^{2}≤ 2}

(c) {1 ,^{1}/_{2},^{1}/_{3},^{1}/_{4},^{1}/_{5}, ...}

(d) {*x*∈**R**|*x*^{2n+1}= 2 for some*n*∈**N**}

(e) {^{m}/_{n}∈**Q**| 0 ≤*m*<*n*}

(f) {^{m}/_{n}∈**Q**| 0 <*m*<*n*with both*m*,*n*odd}

(g) real numbers in (0, 1) whose decimal expansions do not contain the digit 9,

(h) real numbers in (0, 1) whose decimal expansions contain only odd digits. - Show that the real numbers in (0, 1) whose first decimal digit is ≠ 9 form an interval of length
^{9}/_{10}.

Prove that the real numbers in (0, 1) whose first and second decimal digits are ≠ 9 form an union of nine intervals of total length^{81}/_{100}.

Show that the real numbers with no 9 in the first three digits form a union of intervals of total length (^{9}/_{10})^{3}.Hence prove that the set of Question 3(g) may be enclosed in a union of intervals of arbitrarily small total length. (Such sets are said to have

*measure zero*.)

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