Logic question

[kikiandkoko]

The students of a class (with more than three students) altogether speak four languages. For any three students, there is at least one language that all of them speak. There is no language that all students speak. Prove that each student speaks at least three languages. Construct an example where this happens.
Please help!!!

 

[Dr.Peterson]

Is this just a random puzzle you were given, or is it related to a class you are taking? In the latter case, what have you learned that might be useful here? (This can help us think about what sort of method might be suitable for you. It might also help us think of anything that might be useful to us!)

One thought I have is to try proving by contradiction. Suppose there is a student who speaks only two languages; can we show that this is impossible without contradicting the givens?

You might also just start by trying to create an example, and hope that that will help you see the implications of the givens.