Given two sets $A$ and $B$ with the property that their intersection is empty ($A \cap B = \emptyset$), we want to analyze the expression $|(A \cup B)^n|$ and determine if it is always greater than or equal to $|A^n| + |B^n|$ for every integer $n \geq 1$.

Step-by-Step Analysis

1. Understanding $(A \cup B)^n$:

   • The set $(A \cup B)^n$ represents the set of all $n$-tuples where each element in the tuple is chosen from $A \cup B$.
   • Since $A \cap B = \emptyset$, each element in any $n$-tuple in $(A \cup B)^n$ can independently come from either $A$ or $B$, but not both simultaneously.

2. Cardinality of $(A \cup B)^n$:

   • If we denote $|A| = a$ and $|B| = b$, then $|A \cup B| = a + b$ (because $A$ and $B$ are disjoint).
   • The set $(A \cup B)^n$ then contains $(a + b)^n$ elements, as each position in an $n$-tuple has $a + b$ choices.

3. Cardinality of $A^n$ and $B^n$:

   • The set $A^n$ consists of all $n$-tuples where each element is from $A$ only. Thus, $|A^n| = a^n$.
   • Similarly, $B^n$ consists of all $n$-tuples where each element is from $B$ only. Therefore, $|B^n| = b^n$.

4. Inequality Check:

   • We now compare $|(A \cup B)^n| = (a + b)^n$ with $|A^n| + |B^n| = a^n + b^n$.
   • By the binomial theorem, we have: $(a + b)^n = a^n + b^n + \text{(positive terms involving products of } a \text{ and } b\text{)}.$
   • Since each of the additional terms in the binomial expansion is positive for $n \geq 1$, it follows that: $(a + b)^n \geq a^n + b^n.$

5. Conclusion:

   • Therefore, for every integer $n \geq 1$, the inequality $|(A \cup B)^n| \geq |A^n| + |B^n|$ holds true.

This completes the proof.

Would you like a deeper explanation of any part?

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1. What if $A \cap B \neq \emptyset$? How does it affect the result?
2. What happens to the inequality if $n = 0$?
3. Can this approach be extended if $A$ and $B$ are not finite?
4. How does this relate to combinatorial counting principles?
5. Could this inequality be applied to probability spaces as well?

Tip: When tackling inequalities, consider expanding terms using known theorems (like the binomial theorem) to reveal any hidden positive terms that might support your argument.