# Solve Math Puzzle: Proving Nonemptiness of Intersection

• tauon
In summary, the author is trying to solve a math problem but is confused. He notes that the family of sets that he is trying to include as part of the intersection might not be distinct, and that this can be fixed easily.
tauon
Here's a math problem that's giving me a head ache (though to some of you it might seem to be quite trivial)

$$r,s\in N^\ast$$ , $$r+1\leq s$$ ; $$|A_i|=r, \forall i\in \{1,2,...,s\}$$
the intersection of any $$r+1$$ of sets $$A_i$$ is nonempty [1]

prove that
$$\bigcap_{i=1,s} A_i \neq \emptyset$$ [C]
_______________________________
At first I thought about it like this:

$$\bigcap_{i=1,s} A_i = \emptyset$$
we can arbitrarily chose an $$A_k =\{x_1 , ..., x_r\}$$ then $$\exists i_1, ... , i_r \in \{1, ... ,s\}$$ so that $$x_1 \not\in A_{i_1}, ... , x_r \not\in A_{i_r}$$
$$\Rightarrow A_k\cap A_{i_1} \cap ... \cap A_{i_r} = \emptyset$$ (dem: if the contrary is true and the intersection is nonempty then at least one element of $$A_k$$ is in all the sets- contradiction with how the $$A_{i_j}$$ family of sets was defined)

So there is a family of $$r+1$$ sets with an empty interesection- contradiction with [1] therefore [C] is true.

But then I thought: wait a minute. That doesn't prove [C], it only "verifies" [1], since the last $$A_s$$ might contain no element from $$\bigcap_{i=1,r+1} A_i$$ so in the end it is possible $$\bigcap_{i=1,s} A_i = \emptyset$$ for some arbitrary $$A_k$$ thus constructed.

And now I'm confused and I don't know if I solved the problem right or not... what part am I doing wrong here?

ps: sorry for the messy post, I'm a beginner with the math bbcode.

No your original proof looks right - only note that the i1,...,ir are not necessarily distinct but this is easy to fix - shows why r+1<=s is necessary.

bpet said:
No your original proof looks right - only note that the i1,...,ir are not necessarily distinct but this is easy to fix

oh, yes. I think you're right.

bpet said:
- shows why r+1<=s is necessary.

what do you mean?

edit: never mind, I understand what you meant now. thanks.

Last edited:

## 1. How do you prove the nonemptiness of intersection in a math puzzle?

To prove the nonemptiness of intersection in a math puzzle, you must show that there is at least one element that is common to both sets being intersected. This can be done through various methods such as using logic, algebra, or set theory principles.

## 2. What is the significance of proving nonemptiness of intersection in a math puzzle?

The significance of proving nonemptiness of intersection in a math puzzle is that it confirms the existence of a solution to the puzzle. It also helps to narrow down the possible solutions and provides a starting point for further problem-solving.

## 3. Can you provide an example of a math puzzle where proving nonemptiness of intersection is necessary?

One example of a math puzzle where proving nonemptiness of intersection is necessary is the Sudoku puzzle. In order to solve the puzzle, you must prove that there is at least one number that is common to both the row and column it belongs to, as well as the 3x3 grid it is a part of.

## 4. What are some common techniques used to prove nonemptiness of intersection in math puzzles?

Some common techniques used to prove nonemptiness of intersection in math puzzles include the use of Venn diagrams, mathematical induction, and proof by contradiction. Other methods may also involve algebraic manipulation, logical reasoning, and set theory principles.

## 5. Are there any tips for approaching a math puzzle that requires proving nonemptiness of intersection?

Some tips for approaching a math puzzle that requires proving nonemptiness of intersection include breaking down the problem into smaller parts, using visual aids such as diagrams or tables, and trying out different strategies or techniques until a solution is found. It is also important to carefully read and understand the given information before attempting to solve the puzzle.

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