# Thread: How many different languages?

1. ## How many different languages?

A company specializing in international trade has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. At least how many different languages are spoken by the employees of this company?

I think the answer is 70 (the case where each person speaks only one language - that nobody else speaks). Am I right?

2. ## Re: How many different languages?

Originally Posted by alexmahone
A company specializing in international trade has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. At least how many different languages are spoken by the employees of this company?

I think the answer is 70 (the case where each person speaks only one language - that nobody else speaks). Am I right?
70 is a possible answer but not the least possible. 70 would be the minimum possible answer if one employee speaks only one language.

EDIT: for example suppose the company only had 3 employees A,B and C. Then only 2 languages
k and m would suffice. A speaks k and m both. B speaks only k and C speaks only n.

3. ## Re: How many different languages?

But then there does not exist a language that B or C speaks that A does not. In your example B speaks k and so does A which does not satisfy "and also a language that B speaks but A does not".

4. ## Re: How many different languages?

Originally Posted by alexmahone
A company specializing in international trade has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. At least how many different languages are spoken by the employees of this company?

I think the answer is 70 (the case where each person speaks only one language - that nobody else speaks). Am I right?
Now afgter some thought i think 7 languages will suffice. Let $L$ is a seven element subset where each element of $L$ represents a language. there are $2^7-1=127$ proper subsets of a 7 element set. Choose any 70 subsets of $L$ and assign each subset to an employee.

5. ## Re: How many different languages?

Originally Posted by alexmahone
A company specializing in international trade has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. At least how many different languages are spoken by the employees of this company?

I think the answer is 70 (the case where each person speaks only one language - that nobody else speaks). Am I right?

The requirement on A and B is equivalent to the statement that the sets of A's and B's languages are non-empty and neither is a subset of the other. Let there be 8 languages and assign each subset of size 4 to an employee. There are C(8,4) = 70 such subsets, and no subset of size 4 can be a subset of another subset of size 4.

That at least 8 languages are required is a consequence of Sperner's Theorem:

If $A_1, A_2, \dots , A_m$ are subsets of $N = \{1, 2, \dots , n\}$ such that $A_i$ is not a subset of $A_j$ if $i \neq j$, then $m \leq \binom{n}{\lfloor n/2 \rfloor}$.

6. ## Re: How many different languages?

It can't be 8. Any repetition of a language at any point would allow you to find two employees A and B that both speak a common language. At least 70 must be spoken.

Has the world gone mad or am I misunderstanding the question?

7. ## Re: How many different languages?

Originally Posted by terrorsquid
It can't be 8. Any repetition of a language at any point would allow you to find two employees A and B that both speak a common language. At least 70 must be spoken.

Has the world gone mad or am I misunderstanding the question?
suppose A speaks greek and latin and B speaks Latin and English. Then there exists a language (greek) which A speaks but B does not and there exists a language (English) which B speaks but A does not. Although there is a language(latin) which is shared by A and B still this doesn't violate the conditions of the question.

8. ## Re: How many different languages?

Originally Posted by abhishekkgp
suppose A speaks greek and latin and B speaks Latin and English. Then there exists a language (greek) which A speaks but B does not and there exists a language (English) which B speaks but A does not. Although there is a language(latin) which is shared by A and B still this doesn't violate the conditions of the question.
Ah, ok. That makes more sense. I read it as A can't know a language that B knows etc. thanks. Although that is 3 languages for 2 people :S

Your other example doesn't make sense though as B doesn't speak a language that A does not (and neither does C assuming m=n).

So, an example of n people speaking less than n languages while satisfying the question would be:

A(1,2), B(2,3), C(3,4), D(4,1), E(4,2), F(1,3)

It seems you can't do it for less than 4 languages.

9. ## Re: How many different languages?

Originally Posted by terrorsquid
Ah, ok. That makes more sense. I read it as A can't know a language that B knows etc. thanks. Although that is 3 languages for 2 people :S

Your other example doesn't make sense though as B doesn't speak a language that A does not (and neither does C assuming m=n).

So, an example of n people speaking less than n languages while satisfying the question would be:

A(1,2), B(2,3), C(3,4), D(4,1), E(4,2), F(1,3)

It seems you can't do it for less than 4 languages.
yes i had realized my mistake. thanks anyway!!