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Let L = {+, · P} where + and · are binary function symbols and P is a unary predicate symbol, and let M be an L-structure where its domain |M| is the set R of real numbers, + and · are the usual addition and multiplication over R respectively, and P = {r ∈ R | r > 0}.

(i) Give an L-formula ϕ1(x) that defines the set {1} ⊂ R in M.

(ii) Give an L-formula ϕ2(x, y) that defines the less-than relation < over R in M.

(iii) Give an L-formula ϕ3(x) that defines the set {√2} ⊂ R in M.

(iv) Give an L-formula ϕ4(x) that defines the interval [1,√2) ⊂ R in M.

I've got the following, after a lot of struggling, but I'm unsure as to whether they're correct / I'm allowed to plug in the formulae into one another as follows:

φ1(x): ∀y(x⋅y = y)

φ2(x,y): ∃z(P(z) ∧ (x+z = y))

φ3(x): ∀y(φ1(y) → ((y+y) = (x⋅x))) ∧ P(x)

φ4(x): (∀y∀z(φ3(y) ∧ φ2(x,y) ∧ φ1(z) ∧ φ2(z,y))) ∨ φ1(x)

Please correct me if I'm wrong! :)

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  • $\begingroup$ Welcome to Math.SE! You might want to take a look at the notation help page to see how to write Math at this site. $\endgroup$ – AlexR May 26 '15 at 11:34
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$\varphi_4$ is false, the rest is okay. $\varphi_4$ only defines $1$. The proper solution would be

$$\varphi_4(x) : (\forall y\ \forall z ((\varphi_3(y) \wedge \varphi_1(z))\implies (\varphi_2(z,x) \wedge \varphi_2(x,y)))) \vee \phi_1(x)$$

More concise: $$x\in [1,\sqrt 2) \Leftrightarrow x=1 \vee (1 < x \wedge x < \sqrt 2)$$

We translate $1<x$ to $\forall z(\varphi_1(z) \implies \varphi_2(1,x))$ and similarly for $x<\sqrt 2$.

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