The question asks whether the following statement is true or false:

"For all sets $A$ and $B$, either $A \subseteq B$ or $B \subseteq A$."

This statement is false.

Explanation:

• The statement suggests that for any two sets $A$ and $B$, one must always be a subset of the other. However, this is not true in general. Consider two sets that have some elements in common but are not subsets of each other. For example:

  Let $A = \{1, 2\}$ and $B = \{2, 3\}$.

  • $A \subseteq B$ is false because $A$ contains $1$, which is not in $B$.
  • $B \subseteq A$ is false because $B$ contains $3$, which is not in $A$.

Thus, there exist sets for which neither is a subset of the other, proving the statement false.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the difference between a subset and a proper subset?
2. Can two sets be subsets of each other without being identical?
3. How can the Venn diagram illustrate relationships between subsets?
4. What are examples of disjoint sets?
5. How do we define the power set of a set?

Tip: Always consider counterexamples when evaluating the truth of a universal statement in set theory.