# English to Predicate Logic (Imply and AND)

The question is:

If Bob is happy, then all his friends are happy

My attempt looks like:

$happy(bob) \Rightarrow (\forall x(friend(x, bob) \wedge happy(x)))$

$happy(bob) \Rightarrow (\forall x(friend(x, bob) \color{red}{\Rightarrow} happy(x)))$

So is my answer acceptable too? If not why?

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If Bob is happy, then { for all $x$: $x$ is friends with Bob and $x$ is happy }

whereas what you want is

If Bob is happy, then { for all $x$: if $x$ is Bob's friend then $x$ is happy }

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Oh so I might be missing out if some friends of bob are unhappy? – Jiew Meng Sep 28 '12 at 13:18
No - your answer says that everyone is friends with Bob. You need to say that people are happy if they are friends with Bob (which is why you need the implication). If someone is not friends with Bob, then we don't know if they're happy or unhappy. – Chris Taylor Sep 28 '12 at 13:27