Let's fill in the blanks systematically to complete the statement:

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Let $A$ and $B$ be nonempty sets. We say that $A$ and $B$ have the same cardinality if there exists a bijective function $f: A \to B$. We say that the cardinality of $A$ is $n$ if there is a bijective function from $A$ to $\{1, 2, \ldots, n\}$. If such a function exists, we call $A$ finite. If $A$ is finite, we denote the cardinality of $A$ by $\#A$.

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Breakdown:

1. Cardinality: This term refers to the size or number of elements in a set. Two sets are said to have the same cardinality if there is a one-to-one and onto (bijective) function between them.

2. Bijective Function: This is a function that is both injective (one-to-one) and surjective (onto). It's crucial for showing that two sets have the same number of elements.

3. Finite Sets: A set is finite if it has a bijection with a set of the form $\{1, 2, \ldots, n\}$, where $n$ is a positive integer.

4. Notation $\#A$: This is a way to represent the number of elements in a finite set $A$.

Would you like to explore this further or have any questions?

Related Questions:

1. What is the cardinality of the set $\{a, b, c\}$?
2. How do we define the cardinality of an infinite set?
3. What is the difference between injective, surjective, and bijective functions?
4. How would you prove that two sets have the same cardinality?
5. What is an example of a finite set and its cardinality?

Tip:

When proving that two sets have the same cardinality, always check if you can construct a bijective function between them.