Mathematics-Computer Science 4215H Mathematical Logic Trent University, Winter 2021Assignment #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. 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 de- termine the possible lengths of formulas of this language. [Do this for the language you chose in Problem 5.2.] 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.  (4) n0 = 1 for every n different from 0 in LN . 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. 5.11. [Problem 5.11] Give a precise definition of the scope of a quantifier. 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.