1. Prove the associative law holds on Q.
Let r,s,t ∈ Q. Prove (rs)t = r(st).
2. Consider the relation ~ on N × N defined by (a,b)~(c,d) if and only if a + d = b + c.
~ is an equivalence relation. (T/F)
3. Let b ∈ R. There exists a ∈ N such that b < a < b + 1.(T/F)
4. Consider again the relation ~ on N × N defined by (a,b)~(c,d) if and only if a+d=b+c. The set of equivalence classes of ~ gives us the set of rationals Q.(T/F)
5. Let a ∈ N with a ≠ 1. There exist at least two elements b1,b2 ∈ N such that s(b1) = s(b2)= a.(T/F)
6. (x + y)z = x(y + z) for all x, y, z ∈ Z.(T/F)
7. Define function i: Z → Q by i(x) = [(x,1)] for all x ∈ Z. This function is injective.(T/F)
8. Which of the following sets is a Dedekind cut (highlight all that apply):
(2/3,∞)∩Q
N
Q