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Question 6.10: Give a model of the sentence φeq = ∀x [R1(x, x)] ∧∀x,y [R1(x...

Give a model of the sentence

ϕeq=∀x[R1(x,x)]\phi _{eq} = \forall x [R_{1}(x,x) ]
∧∀x,y[R1(x,x)⟷R1(y,x)]\wedge \forall x,y [R_{1}(x,x)\longleftrightarrow R_{1}(y,x)]
∧∀x,y,z[R1(x,y)∧R1(y,z)→R1(x,z)]\wedge \forall x,y,z [R_{1}(x,y)\wedge R_{1}(y,z) \rightarrow R_{1}(x,z)].

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The statement ϕeq\phi _{eq} gives the three conditions of an equivalence relation. A model (A, R1R_{1}), where A is any universe and R1R_{1} is any equivalence relation over A, is a model of ϕeq\phi _{eq}. For example, let A be the integers Z and let R1R_{1} = {(i, i)| i ∈ Z}.

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