# Logical Definition of Two Disjoint Sets

• MushroomPirat
In summary, the conversation discusses the possibility of writing the statement "Sets A and B are disjoint" using symbols other than the given A \cap B = \emptyset. The individual proposes using the logical equivalence notation, and after realizing it is a trivial proof, suggests deleting the thread.
MushroomPirat

## Homework Statement

I'm self-studying Daniel Velleman's How to Prove It, and I'm wondering if there is some way to write that "Sets A and B are disjoint" using symbols, other than the $A \cap B = \emptyset$ given in the book.

## The Attempt at a Solution

I'm thinking that if A and B are disjoint, then for every element x, where if x is an element of A, then x is not an element of B. Or:

$\forall x (x\in A \rightarrow x \notin B)$

Thanks !Edit: Right after I posted this, I realized I could just prove the equivalency, and it turned out to be a pretty trivial proof (if I did it right). So, I think this thread can be deleted now.

Last edited:
A \cap B = \emptyset \begin{aligned}\forall x (x\in A \rightarrow x \notin B) &\iff \\\forall x (x\in A \lor x \notin B) &\iff \\\forall x ((x\in A \land x \notin B) \lor (x \notin A \land x \notin B)) &\iff \\\forall x (x\in A \land x \notin B) &\iff \\\forall x \neg(x\in A \land x \in B) &\iff \\\neg \exists x (x\in A \land x \in B) &\iff \\A \cap B = \emptyset\end{aligned}

## 1. What is the logical definition of two disjoint sets?

The logical definition of two disjoint sets is that they have no elements in common. This means that the intersection of the two sets is an empty set.

## 2. How can you prove that two sets are disjoint?

To prove that two sets are disjoint, you can show that their intersection is an empty set. This can be done by listing out the elements of each set and showing that there are no common elements between them.

## 3. Can two sets be disjoint if they have the same number of elements?

Yes, two sets can be disjoint even if they have the same number of elements. The key factor in determining if sets are disjoint is whether they have any elements in common, not the number of elements they contain.

## 4. Are disjoint sets always mutually exclusive?

Yes, disjoint sets are always mutually exclusive. This means that they cannot have any elements in common, making them completely separate from each other.

## 5. How can understanding disjoint sets be useful in real-life scenarios?

Understanding disjoint sets can be useful in many situations, such as in data analysis, where you may want to compare two different groups and see if they have any common characteristics. It can also be helpful in problem-solving and decision-making, as it allows you to clearly define and differentiate between different groups or categories.

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