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I have a statement.

Every student at your school either can speak Russian or knows C++.

Where P(x) is the statement "x can speak Russian". And Q(x) is the statement is the statement "x knows the computer language C++". I have to express this in terms of logical connectives. Here's what I did.

∀x(P(x) ∨ Q(x))

Is this correct?

Fakhr Ali
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  • https://en.wikipedia.org/wiki/XOR_gate – Dmitry Nov 03 '19 at 15:31
  • either can also means both – Ethan Nov 03 '19 at 15:32
  • @Manx This 'either' is the reason I posted the question. I can't quite figure out how to express it. – Fakhr Ali Nov 03 '19 at 15:36
  • Spoken language, at least English, is ambiguous on this point. If one says "you can have either steak or chicken" the strong implication is that you must choose, you can not have both. If, however, I say "all the students have either a pen or a pencil" I think the strong presumption is that they might have both. Presumably the issue is "can all the students write stuff down?" so having both is still a positive. In your case, I would read the "either" as allowing for "both" but, as I say, it is somewhat ambiguous. – lulu Nov 03 '19 at 15:41

1 Answers1

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Basicly, if either also include both, your answer is correct.

If not, you will need to write $$\forall x,(P(x)\land\neg Q(x))\lor (\neg P(x)\land Q(x))$$

In general, one of $p,q$ is true but not both can be written as $p⊕q$, and have following equivalent forms:

$$p⊕q\equiv (p\land\neg q)\lor (\neg p\land q)\equiv(p\lor q)\land(\neg p\lor\neg q)\tag*{aka Xor}$$

Sometime it's hard to tell if it means both or not, you can also check this post.

Ethan
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