**(A)** (∃x) (boy(x) → (∀y) (girl(y) ∧ taller(x,y))) **(B)** (∃x) (boy(x) ∧ (∀y) (girl(y) ∧ taller(x,y))) **(C)** (∃x) (boy(x) → (∀y) (girl(y) → taller(x,y))) **(D)** (∃x) (boy(x) ∧ (∀y) (girl(y) → taller(x,y)))

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

Some boys in the class are taller than all the girls

Note: $\text{taller} (x, y)$ is true if $x$ is taller than $y$.

- $(\exists x) (\text{boy}(x) \rightarrow (\forall y) (\text{girl}(y) \land \text{taller}(x, y)))$
- $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \land \text{taller}(x, y)))$
- $(\exists x) (\text{boy}(x) \rightarrow (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$
- $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$

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179 votes

Best answer

Now many people get confused when to use $\wedge$ and when to use $\implies$. This question tests exactly that.

We use $\wedge$ when we want to say that the both predicates in this statement are always true, no matter what the value of $x$ is. We use $\implies$ when we want to say that although there is no need for left predicate to be true always, but whenever it becomes true, right predicate must also be true.

Now we have been given the statement $\text{ “Some boys in the class are taller than all the girls"}$. Now we know for sure that there is at least a boy in class. So we want to proceed with $ “\left(\exists x\right)(boy\left(x\right)\wedge"$ and not $ “\left(\exists x\right) (boy\left(x\right) \implies"$, because latter would have meant that we are putting no restriction on the existence of boy i.e. there may be a boy-less class, which is clearly we don't want, because in the statement itself, we are given that there are some boys in the class. So options (A) and (C) are ruled out.

Now if we see option (B), it says, every y in class is a girl i.e. every person in class is a girl, which is clearly false. So we eliminate this option also, and we get correct option (D). Let us see option (D)explicitly also whether it is true or not. So it says that if person y is a girl, then x is taller than y, which is really we wanted to say.

So option (**D**) is correct.

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21 votes

I have attached two links here. They contain photos clicked from my worked out solution. Hopefully this will clear the doubts arising between option B and D.

15 votes

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5 votes

we have been given the statement "Some boys in the class are taller than all the girls". Now we know for sure that there is atleast a boy in class. So we want to proceed with "(∃x) (boy(x) ∧" and not "(∃x) (boy(x) →", because latter would have meant that we are putting no restriction on the existence of boy i.e. there may be a boy-less class, which is clearly we don't want, because in the statement itself, we are given that there are some boys in the class. So options (A) and (C) are ruled out.

Now if we see option (B), it says, every y in class is a girl i.e. every person in class is a girl, which is clearly false. So we eliminate this option also, and we get correct option (D). So it says that if person y is a girl, then x is taller than y, which is really we wanted to say.

So option (D) is correct.

Now if we see option (B), it says, every y in class is a girl i.e. every person in class is a girl, which is clearly false. So we eliminate this option also, and we get correct option (D). So it says that if person y is a girl, then x is taller than y, which is really we wanted to say.

So option (D) is correct.