Question 1: Complete the proof of Lemma, that is, prove that the relation 
is reflexive and symmetric.
Question 2: Complete the proof of Lemma, that is, prove that the binary operation + , the unary operation-1 and the relation <, all on Q, are well defined.
Question 3: Let x ε Z and y ε Z*.
a) Prove that [(x,y)] = 0‾ if and only if x = 0.
b) Prove that [(x,y)] = 1‾ if and only if x = y.
c) Prove that 0 < [(x,y)], if and only if 0 < xy.
Question 4: Prove the theorem:
Let r, s, t ε Q.
1. (r + s) + t = r + (s + t) (Associative Law for Addition).
2. r + s = s + r (Commutative Law for Addition).
3. r + 0‾ = r (Identity Law for Addition).
4. r+ (-r) = 0‾ (Inverses Law for Addition).
5. (rs)t = r(st) (Associative Law for Multiplication).
6. rs = sr (Commutative Law for Multiplication).
7. r • 1‾ = r (Identity Law for Multiplication).
8. If r ≠ 0‾, then r • r-1 = 1‾ (Inverses Law for Multiplication).
9. r(s + t) = rs + rt (Distributive Law).
10. Precisely one of r < s or r = s or r > s holds (Trichotomy Law).
11. If r < s and s < t, then r < t (Transitive Law).
12. If r < s then r + t < s+t (Addition Law for Order).
13. If r < s and t > 0, then rt < st (Multiplication Law for Order).
14. 0‾ ≠ 1‾ (Non-Triviality).
Question 5:
Let i: Z → Q be defined by i(x) = [(x, 1)] for all x ε Z
1. The function i: Z → Q is infective.
2. i(0) = 0‾ and i(1) = 1‾.
3. Let x,y ε Z. Then
a. i(x +y) = i(x) +i(y);
b. i(-x) = -i(x);
c. i(xy) = i(x)i(y);
d. x < y if and only if i(x) < i(y).
4. For each r ε Q there are x,y ε Z such that y ≠ 0 and r i(x)(i(y))-1
Question 6:
Let r, s, p, q ε Q.
(1) Prove that —1 < 0 < 1.
(2) Prove that if r < s then —s < —r.
(3) Prove that r . 0 = 0.
(4) Prove that if r > 0 and s > 0, then r + s > 0 and rs > 0.
(5) Prove that if r > 0, then 1/r > 0.
(6) Prove that if 0 < r < s, then 1/s < 1/r
(7) Prove that if 0 < r < p and 0 < s < q, then rs < pq.
Question 7:
1) Prove that 1 < 2
2) Let s,t ε Q. Suppose that s < t. Prove that (s+t)/2 ε Q, and that s < (s+t)/2 < t.
Question 8: Let r ε Q. Suppose that r > 0.
(1) Prove that if r = a/b for some a,b ε Z, such that b ≠ 0, then either a > 0 and b > 0, or a < 0 and b < 0.