Which of the following are terms of one of the languages defined in Examples 5.1 and 5.2? If so, which of these language(s) are they terms of; if not, why not? (1) ·v2 (3) |1 + v30 (5) + + · + 00000 [1.5 = 3×0.5 each]

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Mathematics-Computer Science 4215H – Mathematical Logic
Trent University, Winter 2021
Assignment #6
Due on Friday, 5 March.
Do all of the following problems, all of which are straight out of the textbook0
(which
explains the numbering), reproduced here for your convenience.
5.1. [Problem 5.1] Which of the following are terms of one of the languages defined in
Examples 5.1 and 5.2? If so, which of these language(s) are they terms of; if not, why
not?
(1) ·v2 (3) |1 + v30 (5) + + · + 00000 [1.5 = 3×0.5 each]
5.2. [Problem 5.2] Choose one of the languages defined in Examples 5.1 and 5.2 which has
terms of length greater than one and determine the possible lengths of terms of this
language. [2]
5.4. [Problem 5.4] Which of the following are formulas of one of the languages defined in
Examples 5.1 and 5.2? If so, which of these language(s) are they formulas of; if not,
why not?
(1) = 0 + v7 · 1v3 (3) (|v20 → ·01) (5) < +01|v1v3 [1.5 = 3×0.5 each]
5.6. [Problem 5.6] Choose one of the languages defined in Examples 5.1 and 5.2 and determine the possible lengths of formulas of this language. [Do this for the language
you chose in Problem 5.2.] [2]
5.8. [Problem 5.8] In each case, write down a formula of the given language expressing the
given informal statement.
(2) “There is an empty set” in LS. [1]
(4) “n
0 = 1 for every n different from 0” in LN . [1]
5.9. [Problem 5.9] Define first-order languages to deal with the following structures and,
in each case, an appropriate set of axioms in your language: (2) Groups. [3]
5.11. [Problem 5.11] Give a precise definition of the scope of a quantifier. [2]
5.12. [Problem 5.12] Which of the formulas you gave in solving Problem 5.8 are sentences?
[1 = 2×0.5 each]
[Total = 15]
0 A Problem Course in Mathematical Logic, Version 1.6