MATH 3210 § 1 FOURTH HOMEWORK ASSIGNMENT  Due Friday,
A. Treibergs    September 19, 2008.

A. Please hand in the following problems from Taylor's Foundations of Analysis.
  • 20 [ 7 ]
B. Please hand in the following additional problem.
  1. In a commutative ring (R, +, ×), show that for all x ∈ R, -(-x) = x.

  2. Assume that the integers, Z, satisfiy the axioms of a commutative ring. Show that the construction of the rational numbers (Q, +, ×), given on pp. 17-18 satisfies the distributive axiom "D" on p. 16.

  3. Suppose that the relation ≡ on the integers Z is given for x, y ∈ Z by

                x ≡ y   ⇔   ( ∃ z ∈ Z ) ( x = y + 7z ).

    1. Show that ≡ is an equivalence relation.
    2. Show: the binary operation given for equivalence classes â, û &isin Z/≡ by

                 â # û := (a + u)^

      is well defined. (That is, if we used different numbers in the equivalence classes b ≡ a and v ≡ u, then the answer is still the same equivalence class: a + u ≡ b + v.)