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I am working through Velleman's "How to Prove It", and I have some uncertainties about this exercise.

Analyze the logical forms of the following statements:

  1. Either both Ralph and Ed are tall, or both of them are handsome.
  2. Both Ralph and Ed are either tall or handsome
  3. Both Ralph and Ed are neither tall nor handsome
  4. Neither Ralph nor Ed is both tall and handsome

Here are my solutions:

  • Let $R_T =$ Ralph is tall, $R_H =$ Ralph is handsome, $E_T =$ Ed is tall, and $E_H =$ Ed is handsome.

    1. $(R_T \land E_T) \lor (R_H \land E_H)$
    2. $(R_T \lor R_H) \land (E_T \lor E_H)$
    3. $(\neg R_T \land \neg R_H) \land (\neg E_T \land \neg E_H)$
    4. $\neg (R_T \land R_H) \land \neg (E_T \land E_H)$

For #4 is it correct to interpret "neither Ralph nor Ed" as the conjunction of the two other negated conjunctions?

Thank you in advance!

Iyeeke
  • 962

1 Answers1

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For number 3, you it can help to split up the sentence. For example, another way of expressing it would be "Ralph is neither tall nor handsome, and Ed is neither tall nor handsome". Considering this, then:

$ 3. ¬(RT∨RH)∧¬(ET∨EH) $

For number 4, we can do something similar. We can reexpress it as "Neither Ralph is both tall and handsome, nor Ed is both tall and handsome". Hence:

$ 4. ¬(RT∧¬RH)∨¬(ET∧EH) $

Alex Ruiz
  • 318