Translate the statements into set inclusion

• MHB
• mathmari
In summary: Also, from (vi) we know that mathematicians are scientists and from (ii) we know that scientists are human. So the drowned mathematician must also be human, right? 🤔Therefore, we can conclude that there was a mathematician who was not a swimmer. 🎉In summary, we can deduce from the given statements that there was a mathematician who was not a swimmer. This is based on the fact that the mathematician was drowned (v), which means they were not a swimmer (i). Also, from (vi) and (ii), we know that the mathematician was a scientist and a human. Therefore, the statement "There was a mathematician who was not
mathmari
Gold Member
MHB
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? $$\exists x\in M : x\in S^c$$

mathmari said:
(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?

Klaas van Aarsen said:
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:
Klaas van Aarsen said:
Yes, but that was not the question was it?

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

mathmari said:
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?

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

mathmari said:
(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?

1. What is set inclusion in translation statements?

Set inclusion in translation statements refers to the relationship between two sets, where one set is completely contained within the other. This means that every element in the first set is also a member of the second set.

2. How do you represent set inclusion in translation statements?

Set inclusion is represented using the subset symbol, ⊆, which is read as "is a subset of". The statement A ⊆ B means that every element in set A is also a member of set B.

3. What is the difference between set inclusion and set equality?

Set inclusion refers to the relationship between two sets where one set is completely contained within the other. Set equality, on the other hand, refers to the relationship between two sets where they have exactly the same elements. In other words, set equality means that both sets are subsets of each other.

4. How is set inclusion used in mathematical proofs?

Set inclusion is often used in mathematical proofs to show that a statement or equation is true for all elements in a set. By showing that one set is a subset of another, it can be proven that the same is true for all elements in the first set.

5. Can set inclusion be used to compare infinite sets?

Yes, set inclusion can be used to compare infinite sets. For example, the set of natural numbers (N) is a subset of the set of real numbers (R), as every natural number is also a real number. This holds true even though both sets are infinite.

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