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Let us define \begin{align} m &: \mbox{ Maria} \\ s &: \mbox{ Maria's son} \\ C &: \mbox{ works in the city} \\ B &: \mbox{ rides a bicycle} \\ F &: \mbox{ is a chicken farmer}. \end{align}

Then how to symbolize the following two statements?

  1. If no-one working in the city rides a bicycle then Maria doesn't work in the city and her son is not a chicken farmer.

  2. No chicken farmers work in the city and ride a bicycle.

My Attempt:

1.

The statement "no-one working in the city rides a bicycle" can be rephrased as

For every x, if $x$ works in the city then $x$ doesn't ride a bicycle.

This can be symbolized as $$ \forall x \left[ C(x) \rightarrow \overline{B(x)}\right]. $$

So the given statement can be symbolized as $$ \left[ \forall x \left\{ C(x) \rightarrow \overline{B(x)} \right\} \right] \rightarrow \left[ \overline{C(m)} \land \overline{F(s)} \right]. $$

2.

The given statement can be rephrased as

For every $x$, if $x$ is a chicken farmer then it is not the case that $x$ works in the city and rides a bicycle.

So the given statement can be symbolized as $$ \forall x \left[ F(x) \rightarrow \left\{ \overline{C(x) \land B(x)} \right\} \right]. $$

Are my solutions correct?

Saaqib Mahmood
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  • I like that question ! An upvote ! – Spectre Oct 18 '20 at 06:50
  • I think it's correct. –  Oct 18 '20 at 06:52
  • It is ok. Note that in set notation you can formulate $\forall x[P(x)\to\overline{Q(x)}]$ as $P\cap Q=\varnothing$. Thus 2. is just $F\cap C\cap B=\varnothing$. For more examples of quantifiers/natural language transcription you can look at https://math.stackexchange.com/a/2152552/399263 – zwim Oct 18 '20 at 07:32

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