# How to write this quantified logical expression correctly?

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?

• This is a definition or a theorem? – Fabio Lucchini Oct 13 '16 at 9:55
• Just an excercise to teach us writing logical expressions with quantifiers. – Accelerate to the Infinity Oct 13 '16 at 9:57

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.