Define a Dedekind cut and show the given set is(n't) a Dedekind cut

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A subset L of the rational numbers is defined as a Dedekind cut if it is proper, has no maximal element, and satisfies a specific property regarding its elements. The set P, defined as {x^4 | x ∈ L}, is determined to be a proper set with no maximal element but fails to meet the third property, thus it is not a Dedekind cut. Conversely, the set M, defined as {23x | x ∈ L}, is confirmed to be a proper set, has no maximal element, and satisfies all three properties, making it a Dedekind cut. However, there are noted inaccuracies in the reasoning for both P and M, particularly regarding the maximal element claims and the phrasing of the third property. Overall, M qualifies as a Dedekind cut while P does not.
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Homework Statement
a) What is a Dedekind cut? Give the precise definition.

b) Let ##L## be a Dedekind cut. Is ##P=\{x^4|x\in L\}## a Dedekind cut?

c) Let ##L## be a Dedekind cut. Is ##P=\{23x|x\in L\}## a Dedekind cut?
Relevant Equations
##\mathbb{Q}##
a) a subset ##L\subset \mathbb{Q}## is a Dedekind cut if ##L## is proper, ##L## has no maximal element, and
$$\forall a,b\in \mathbb{Q}, [(a<b)\land( b\in L)\Longrightarrow a\in L]$$

b)
Is ##P=\{x^4|x\in L\}## a Dedekind cut?

P is proper:
$$(a\in L)\Longrightarrow (a^4\in P)\Longrightarrow P\neq \emptyset$$ $$(-1\in \mathbb{Q})\land (-1\notin P)\Longrightarrow P\neq \mathbb{Q}$$ So ##P## is a proper set.

P has no maximal element:
$$(\forall x \in L, x<d)\Longrightarrow( \forall x^4 \in P, x^4<d^4)$$ So ##P## has no maximal element.

P does not satisfy the 3rd property:
$$(-1\in \mathbb{Q})\land (a^4 \in P)\land(-1<a^4)\land (-1\notin P)$$ So ##P## is not a Dedekind cut.c)
Is ##M=\{23x|x\in L\}## a Dedekind cut?

M is proper:
$$(x\in L)\Longrightarrow (23x \in M)\Longrightarrow M\neq \emptyset$$ $$(L\neq \mathbb{Q})\Longrightarrow (d\notin L\quad \text{for some}\quad d\in\mathbb{Q})$$ $$\Longrightarrow( \mathbb{Q}\ni 23d\notin M)\Longrightarrow (M\neq \mathbb{Q})$$ So ##M## is a proper set.

M has no maximal element:
$$(\forall x\in L, x<d)\Longrightarrow (\forall 23x \in P, 23x<23d)$$ So ##M## has no maximal element.

M satisfies the 3rd property:
##L## satisfies the 3rd property, so $$\forall a,b\in \mathbb{Q}, [(a<b)\land( b\in L)\Longrightarrow a\in L]$$ Multiplying the ##a##s and ##b##s by ##23## leads to $$\forall 23a, 23b\in \mathbb{Q}, [(23a<23b)\land (23b\in M) \Longrightarrow 23a\in M]$$
##M## is a Dedekind cut.
 
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You received no response since you asked no question, but if you want checking of your work, then in b, the argument for no maximal element contains a false claim, i.e. one can. have a<b but b^4< a^4, if a and b are negative.

In part c, the argument that the 3rd property holds is phrased incorrectly. namely one should assume that b is in L and a < 23b, then show a is in M.
 
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