MHB Translate the statements into set inclusion

[mathmari]

Hey!

I am looking at the following:

translate the following statements into set inclusion.
(i) Those who drown are not a fish or a swimmer.
(ii) Scientists are human.
(iii) A person who is not a swimmer is a non-swimmer.
(iv) Fish are not human.
(v) There was a case of a drowned mathematician.
(vi) Mathematicians are scientists.

Check if from the statements (i)–(vi)
,,There was a mathematician who was not a swimmer”
can be implied.
I have done the following:

We consider the sets:
E =Set of drowning, F = Set of Fish, S = Set of swimmers, N = Scientists, H = Human, M = Mathematiker

We have then the following:
(i) $x\in E\rightarrow x\notin (F\cup S)$
(ii) $N\subseteq H$ i.e. $x\in N\rightarrow x\in H$
(iii) $x\in H : x\notin S \rightarrow x\in S^c$
(iv) $F\not\subseteq H$
(v) $\exists x \in (E\cap M)$
(vi) $M\subseteq N$

Is everything correct so far? Could I improve something?

The statement ,,There was a mathematician who was not a swimmer” could be formulated as followes, or not? $$\exists x\in M : x\in S^c$$

 

[I like Serena]

(iv) $F\not\subseteq H$

Hey mathmari!

There can still be a fish that is a human, can't it?

Everything else looks correct to me, although I'm used to seeing $\forall$ in front of them.

The statement ,,There was a mathematician who was not a swimmer” could be formulated as followes, or not? $$\exists x\in M : x\in S^c$$

Yes, but that was not the question was it?

 

[mathmari]

There can still be a fish that is a human, can't it?

Everything else looks correct to me, although I'm used to seeing $\forall$ in front of them.

So we have the following:
(i) $\forall x\in E\rightarrow x\notin (F\cup S)$
(ii) $N\subseteq H$ i.e. $\forall x\in N\rightarrow x\in H$
(iii) $\forall x\in H : x\notin S \rightarrow x\in S^c$
(iv) What do you mean by "There can still be a fish that is a human" ?
(v) $\exists x \in (E\cap M)$
(vi) $M\subseteq N$

Is everything correct except (iv) ? :unsure:

Yes, but that was not the question was it?

Could you give me a hint for that? :unsure:

 

[I like Serena]

So we have the following:
(i) $\forall x\in E\rightarrow x\notin (F\cup S)$
(ii) $N\subseteq H$ i.e. $\forall x\in N\rightarrow x\in H$
(iii) $\forall x\in H : x\notin S \rightarrow x\in S^c$
(iv) What do you mean by "There can still be a fish that is a human" ?
(v) $\exists x \in (E\cap M)$
(vi) $M\subseteq N$

I think it should be:
(i) $x\in E\rightarrow x\notin (F\cup S)$
(ii) $N\subseteq H$ i.e. $x\in N\rightarrow x\in H$
(iii) $\forall x\in H : x\notin S \rightarrow x\in S^c$
(iv) --
(v) $\exists x \in (E\cap M)$
(vi) $M\subseteq N$

As for (iv), you had $F\not\subseteq H$ for "fish are not human".
Suppose $F=\{\text{fish}, \text{human}\}$ and $H=\{\text{human}\}$.
Then $F\not\subseteq H$ is satisfied isn't it? But there is a fish that is a human, which contradicts the desired statement.

Could you give me a hint for that?

We want to deduce that there was a mathematician who was not a swimmer from the given statements.
Perhaps we can begin with (v) that says that there was a case of a drowned mathematician?
Can we apply the other statements to find out that this drowned mathematician was not a swimmer?

 

[mathmari]

We want to deduce that there was a mathematician who was not a swimmer from the given statements.
Perhaps we can begin with (v) that says that there was a case of a drowned mathematician?
Can we apply the other statements to find out that this drowned mathematician was not a swimmer?

(v) There was a case of a drowned mathematician.
(i) Those who drown are not a fish or a swimmer.
(vi) Mathematicians are scientists.
(ii) Scientists are human.
In the case of a fish we have from (iv) that fish are not human. Contradiction.
But we cannot conclude that this drowned mathematician was not a swimmer, right? :unsure:

 

[I like Serena]

I believe (iv) should be $F\cap H=\varnothing$, or $\forall x\in F: x\not\in H$, or $x\in F\to x\not\in H$.

(v) There was a case of a drowned mathematician.
(i) Those who drown are not a fish or a swimmer.
(vi) Mathematicians are scientists.
(ii) Scientists are human.
In the case of a fish we have from (iv) that fish are not human. Contradiction.
But we cannot conclude that this drowned mathematician was not a swimmer, right?

Let's try to split up "Those who drown are not a fish or a swimmer."
It is the same as:
"Those who drown are NOT (a fish OR a swimmer)."
"(Those who drown are NOT a fish) AND (Those who drawn are NOT a swimmer)."
"Those who drown are NOT a fish" and "Those who drawn are NOT a swimmer."

Since there was a mathematician who drowned, we can conclude that they were not a swimmer, can't we?