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Логика на английском №2 / Logic in English #2
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Math 461 (Spring 2020) – Homework 7
The following problems are due on Canvas by Wednesday, April 1, at 11:59pm. Please review the syllabus and Francis Su’s article “Some Guidelines for Good Mathematical Writing” (both
posted on Canvas) for expectations on how to write up your solutions. In each of Problems 2, 3,
and 5, you may write formulas “informally”; just make sure that you could translate
those formulas into the official syntax, if necessary.
1. Consider the formula ϕ given by
(∀v0Ev1 = +v1v0 · v0v1 ∧ ∀v1((¬ < v1v0) ∨ Ev2 < Sv1 + v0v2))
in the language LA (with the binary relation symbol < included). Write the formula (i) “informally”,
and determine all of the (ii) terms, (iii) atomic subformulas, and (iv) free and (v) bound variables.
2. For each of the following notions from mathematics, describe a relevant language L by giving its
constant, function, and relation symbols, and their arities, and provide L-sentences in this language
which give the indicated definition (like we’ve done for graphs and linear/partial orders). You may
have to look up the notion online.
(a) The definition of a group. (Hint: The right definition should involve a binary function.)
(b) The definition of a triangle-free graph.
(c) The definition of f : R → R being a continuous function. (Hint: It should be a version of the
“e/δ-definition”.)
(d) The definition of a (real) vector space.
3. Let Lring be the language of (unital) rings, having constant symbols 0, 1, and binary function
symbols + and ·. Consider the Lring-structures
N = (N, 0, 1, +, ·), Z = (Z, 0, 1, +, ·), Q = (Q, 0, 1, +, ·), R = (R, 0, 1, +, ·), C = (C, 0, 1, +, ·),
where 0, 1, + and · are given their usual interpretations. For each of the 10 pairs M,M′ e
{N , Z, Q, R, C} with M ∕= M′
, give an Lring-sentence ϕ such that M |= ϕ and M′ ∕|= ϕ (or
vice-versa). You may resuse each ϕ for multiple pairs.
4. Let M be an L-structure and t an L-term. Prove that if s1, s2 are variable assignments in M
such that s1(v) = s2(v) for all variables v which occur in t, then s1(t) = s2(t). (Hint: You will need
to use the induction principle for L-terms: Prove that the claim holds when t is a variable, when
t is a constant symbol, and is “closed under” appending a function symbol in front of a bunch of
terms.)
Отвеченные вопросы приведены в содержании.
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