**Problem 41 **: true or false? For any elements a, b and c in an ordered ring R:

(i) Either a < b; or a = b; or a > b

(ii) a < b and b < c imply a < c

(iii) a < b implies a + c < b + c

(iv) a < b and 0 < c imply ac < bc

(v) a < b and c < 0 imply ac > bc

(vi) a ≠ 0; implies a^{2} > 0.
**Ans: all six statements of problem 42 are true. If a, b and c are elements of an ordered field, the "less than" relation has in addition to the statements in problem 42, the following properties:
(vii) 0 < a < b implies 0 < b ^{-1} < a^{-1}
(viii) a < b < 0 implies b^{-1} < a^{-1} < 0.
**

**Problem 42**: suppose a and b are elements of an ordered ring R, prove that |a + b| ≤ |a| + |b|.
**Ans: the absolute value of a is defined as |a| = a if a≥0; and |a| = -a if a≤0. the following basic properties of the absolute value of elements of an ordered ring are derived from this definition:
(1) |a| ≥0; |a| = 0 if and only if a = 0
(2) |ab| = |a||b|; here is the proof:
|ab| = ab by definition. |a| = a and |b| = b, for a and b >0 also by definition. Hence (2).
A similar reasoning is applied to the situation where a or b is <0.
(3) |a + b| ≤ |a| + |b|; here is the proof:
---> -|a| ≤ a ≥|a|; -|b| ≤ b ≥|b|;
sum a and b --> -(|a| +|b|)≤(a + b)≤(|a| + |b|)
Therefore; |a + b| ≤ |a| + |b|. This inequality is known as the Triangle Inequality. We can also arrive at this inequality from (|a + b|)**

**Problem 43**: given a sequence (a_{n}) in an ordered field F, a in F, positive elements ε and n_{0} in F. What does the following math statement mean?

|a_{n} - a| < ε ; whenever n ≥ n_{0}
**Ans: a sequence is basically a list of elements from a given population. Usually (not compulsory) there is a rule or pattern associated with the order of the list; for example, the list (1,2,3,4,5,6,7) is a sequence of +ve integers from 1 to 7. The rule is that each successive number is 1 unit more than the number that precedes it. This sequence is finite, the last element is 7. We can express the sequence of +ve integers as infinite by including three dots after 7, that is, (1,2,3,4,5,6,7...).
The math statement in problem 43 means that the sequence (a _{n}) converges to a, in other words, the limit of the sequuence is a, that is, lim a_{n} = a.
In other words, the interval between a_{n} and a will always be less than the number ε no matter how small we make ε. Which means that no element in the sequence = a. So (a_{n} - a) is called a convergent sequence. **

**Problem 44 **: how is a Cauchy sequence different from a convergent sequence?
**Ans 44: convergent sequence --> |a _{n} - a| < ε whenever n ≥ n_{0}
**

Cauchy sequence --> |a_{n} - a_{m}| < ε whenever m, n ≥ n_{0}

The elements of a Cauchy sequence get close to each other whereas the elements of a convergent sequence get close to a fixed point. A convergent sequence is always a Cauchy sequence but a Cauchy sequence may or may not converge in some ordered fields.

**Problem 45 **: is the real number system R an ordered and complete field? suppose (a_{n}) is a sequence of real numbers, express (a_{n}) as :

(i) a nondrecreasing sequence

(ii) a nonincreasing sequence
**Ans: yes. the real number system R is an ordered and complete field. An ordered field F is complete if every Cauchy sequence of elements in F converges to an element in F.
(i) a _{n} ≤ a_{n+1}
(ii) a_{n} ≥ a_{n+1} **

**Problem 46 **: if n is a +ve integer (n ≠ 0), what is the value of n that produces:

(i) the upper bound of a nonincreasing sequence

(ii) the lower bound of a nondecreasing sequence
**Ans 46: (i) 1; (ii) 1. **

**Problem 47 **: if n is a +ve integer (n ≠ 0),

(i) what is the lower bound of the sequence (a_{n}) = 1/n?

(ii) what is the sum of the lower bound of (a_{n}) and its limit?
**Ans: (i) 0; (ii) try it. **

**Problem 48 **: what does the following math statement mean?

a ≅ b (mod m)
**Ans: the math statement of problem 48 was introduced by the great mathematician Carl Friedrich Guass (04/30/1777 - 02/23/1855). He used it to represent the following definition: two integers a and b shall be said to be congruent for the modulus m when their difference a - b is divisible by the integer m.
When a and b are not congruent, they are called incongruent for the modulus m, that is,
a ≇ b (mod m). **

**Problem 49 **: determine whether or not a and b are congruent modulus m, if a is an even number, b an odd number and m a +ve integer.
**Ans: try it. In general,
a ≅ b (mod m) can be expressed as b = a + km; where k is some multiple of m. **

**Problem 50 **: explain the following basic properties of congruences:

(i)Determination; (ii)Reflexivity; (iii) Symmetry; (iv) Transitivity
**Ans:
(i)Determination implies that either a and b are congruent modulus m or they are not. That is, a ≅ b (mod m) or a ≇ b (mod m)
(ii)Reflexivity implies that a is congruent to itself. That is, a ≅ a (mod m)
(iii)Symmetry implies that when a ≅ b (mod m) then b ≅ a (mod m)
(iv)Transitivity implies that when a ≅ b (mod m); b ≅ c (mod m); then a ≅ c (mod m).**

Peter Oye Sagay