Some boys in the class are taller than all the girls

Question

Identify the correct translation into logical notation of the following assertion.

Some boys in the class are taller than all the girls

Note: taller $(x, y)$ is true if $x$ is taller than $y$.

1. $(∃x)($boy$(x)→(∀y)($girl$(y)∧$taller$(x,y)))$
2. $(∃x)($boy$(x)∧(∀y)($girl$(y)∧$taller$(x,y)))$
3. $(∃x)($boy$(x)→(∀y)($girl$(y)→$taller$(x,y)))$
4. $(∃x)($boy$(x)∧(∀y)($girl$(y)→$taller$(x,y)))$

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My attempt :

Let $x$ is a boy and $y$ is a girl.

So , "all taller girls"

$(∀y)($girl$(y)→$taller$(x,y))$

hence,

"Some boys in the class are taller than all the girls."

$(∃x)($boy$(x)∧(∀y)($girl$(y)→$taller$(x,y)))$

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Can you explain more formally please ?

Answer 1

There isn't really a more formal explanation.   You've translated it correctly.

( Well, except for the line "let x ..."; don't include that.   $x$ isn't "a boy"; $x$ is a token entity in the universe of discussion. )

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"Some boys are something" is : $(\exists x)({\sf boy}(x)\wedge \ldots)$

That something is: "taller than all girls". $(\forall y)\big({\sf girl}(y)\to{\sf taller}(x,y)\big)$

Putting it together: $(\exists x)\Big({\sf boy}(x)\wedge (\forall y)\big({\sf girl}(y)\to {\sf taller}(x,y)\big)\Big)$

That is all.

Answer 2

Explanation of the answer

Continuation of explanation