Table of ContentsQuestion-1............................................................................................................................................2(i)Single closed formula for axioms............................................................................................2(ii)Closed L formulas................................................................................................................2Question 2............................................................................................................................................3(i)First order................................................................................................................................3(ii) Consider the remove (a) and (b)...............................................................................................4Question -3...........................................................................................................................................4(i)L-structure...............................................................................................................................4(ii)b2 =a.....................................................................................................................................5(iii)The L- structure is provide k and c £ k..............................................................................5Question-4............................................................................................................................................5(i)A is countable...........................................................................................................................5(ii)A is an infinite......................................................................................................................6(iii)A∩B is a finite......................................................................................................................6References............................................................................................................................................7
Question-1(i)Single closed formula for axiomsGiven:L → language with qualitym → 2-array function symbole → constant symbolThe single closed formula for axioms group is expressed byLet the l language form is expressed as below:L′G= {∗} and LG= {e, ∗}*→ is a binary function and e → constant symbolL′G- structure are of the form is given byM′=⟨M, +⟩Where + is defined as the binary function on M and the course of the interpretation of * in M′M=⟨M, m, +⟩Let A is a closed formula for LG and L′G is given by(ii)Closed L formulasGiven:A1 = (Q (√ −1) \ {0}; ·)A2 = (Z; +)A3 = (Q; +)A4 is the group of 2 × 2 invertible matrices over Q (under multiplication)A5 is the group of 2 × 2 invertible matrices of determinant 1 over QAi|= φj ⇔ i = j i, j ∈ {1 . . . 5}:Ai│=Φj = Φi↔ i = j , Φi= j={Φ1, Φ2, Φ3, Φ4, ..... Φ5}
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