461_midterm_2_practice_soln.pdf - Math 461(Spring 2018 \u2013 Practice Midterm 2 The following is meant as a practice version of Midterm 2 The real Midterm

# 461_midterm_2_practice_soln.pdf - Math 461(Spring 2018 –...

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Math 461 (Spring 2018) – Practice Midterm 2 The following is meant as a practice version of Midterm 2. The real Midterm 2, on Monday, April 16, in class , will be similar, but shorter. You only need to use the “official” syntax for terms and formulas in the first two problems. 1. Let L be a language consisting of the constant symbol 0 , a binary function symbol +, and a binary relation symbol < . In following formula ϕ : ( v 0 v 1 < 0 + v 0 v 1 → ∃ v 2 ( < 0 + v 2 v 2 ∧ ¬ < v 1 v 2 )) write ϕ out “informally” (without using prefix/Polish notation, adding in parentheses, etc), and identify all of the (i) terms, (ii) atomic formulas, (iii) free variables, and (iv) bound variables. Solution: Informally, we can write this formula as ( x y ( 0 < x + y )) ( z ( 0 < z + z y 6 < z )) The (i) terms in (the formal version of) ϕ are: v 0 , v 1 , v 2 , 0 , + v 0 v 1 , + v 2 v 2 . The (ii) atomic formulas in ϕ are: < 0 + v 0 v 1 , < 0 + v 2 v 2 , < v 1 v 2 . The (iii) free variables in ϕ are: v 1 . The (iv) bound variables in ϕ are: v 0 , v 1 , v 2 . 2. Let L be a language L . (a) Give the generating family for the set of all L -terms, and state the corresponding induction principle (i.e., “induction on L -terms”). (b) Given an L -structure M , and a variable assignment s in M (with domain M ), prove by induc- tion on L -terms that s ( t ) M for all L -terms t . [Typo in old version: It should be s ( t ), not s ( t ).] Solution: (a) The set of L -terms, Term L , is generated by (Exp L , Var ∪ C , {F f } f ∈F ) where Exp L is the set of all L -expressions (i.e., finite sequences of symbols from the language), Var = { v n : n N } is the set of all variables, C is the set of constant symbols in the language, and for each function symbol f ∈ F in the language, say with arity k , F f : Exp k L Exp L is given by F f ( t 1 , . . . , t k ) = ft 1 . . . t k . [the concatenation of these strings of symbols]
The corresponding induction principle is as follows: If S Term L contains all variables and con- stant symbols (i.e., Var ∪C ⊆ S

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