文档介绍：Math 557 – Midterm Exam #2
November 4, 2005
SOLUTIONS
1. Let M = (UM , fM , gM , IM ) where UM = {0, 1, 2, 3, 4}, IM is the identity
relation on UM , and fM , gM are the binary operations of addition and
multiplication modulo 5. Thus M is essentially just the ring of integers
modulo 5. Let L be the language consisting of f, g, I. Note that M is
a normal L-structure.
Write an L-sentence A such that for all normal L-structures M ′, M ′
satisfies A if and only if M ′ is isomorphic to M.
Solution. A brute force solution is to let A be ∃x0 ∃x1 ∃x2 ∃x3 ∃x4 B,
where B is the conjunction of {∀y (Ix0y ∨ Ix1y ∨ Ix2y ∨ Ix3y ∨ Ix4y)}∪
{¬ Ixixj : i =6 j} ∪{Ifxixjxk | i + j = k mod 5} ∪{Igxixjxk | ij = k
mod 5} with i, j, k ranging over 0, 1, 2, 3, 4.
Another solution is to let A be a sentence describing a field consisting
of 5 elements. Namely, let A be the conjunction of the field axioms plus
“there exist exactly 5 things”. We are using the algebraic fact that, up
to isomorphism, there is exactly one field of 5 elements.
2. Let G be a group. For a ∈ G write an = a·· · ··a (n times). We say that
G is torsion-free if for all a ∈ G, if a =6 1 then an =6 1 for all positive
integers n.
(a) Exhibit an infinite set of sentences, S, such that for all groups G,
G is torsion-free if and only if G satisfies S.
Solution. Let S = {An : n ≥ 2}, where An is the sente