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I have to construct a quantified logical expression from statement: "Each circle with center in 0 in complex plane is made of points, which have the same absolute value".

My attempt is the following: $$(\forall \mathcal{K} \in \mathbb{C}, \mathcal{K} = \{z\in \mathbb{C}\mid |z| < r, r \in \mathbb{C}\})((a\in \mathcal{K} \land b \in \mathcal{K}) \implies (|a| = |b|)).$$

Is it correct? Shall I declare what are $a$ and $b$ somewhere?

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  • $\begingroup$ This is a definition or a theorem? $\endgroup$ – Fabio Lucchini Oct 13 '16 at 9:55
  • $\begingroup$ Just an excercise to teach us writing logical expressions with quantifiers. $\endgroup$ – Accelerate to the Infinity Oct 13 '16 at 9:57
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Taking $$\forall K(\text{$K$ is a complex circle with center in $0$}\iff\exists r>0(K=\{z\in\mathbb C:|z|=r\}))$$ as definition, then $$\forall K(\text{$K$ is a complex circle with center in $0$}\implies\forall a\in K\forall b\in K(|a|=|b|))$$ is a theorem.

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