Complete the following:
Let
A
and
B
be nonempty sets.
We say that
A
and
B
have the same  if there exists  function
f
:
A
→
B
.
We say that the  of
A
is
n
if there is  function from
A
to .
If such a function exists we call
A
finite.
If
A
is finite we denote the  of
A
by
#
A
.

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.

Let A and B be nonempty sets. We say that A and B have the same if there exists a function f: A → B. We say that the of A is n if there is a function from A to . If such a function exists we call A finite. If A is finite we denote the of A by #A.

Explore the concept of cardinality in set theory and learn how to define the cardinality of finite sets through bijective functions. This problem explains key ideas such as bijections, finite sets, and how to calculate the cardinality using the #A notation. Suitable for undergraduate students studying discrete mathematics or set theory.

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