Complete the following:

Let
A
and
B
be sets.

The selectintersectionuniondifferencesumCartesian productcomplement of the sets
A
and
B
, denoted by
A
×
B
, is the set of all selectunordered pairsordered pairssetsnumbers
(
a
,
b
)
where
a
is selectrelated toan element ofa subsets ofnot related tonot an element ofa proper subset ofequal tonot equal to the set
A
selectandor
b
is selectrelated toan element ofa subsets ofnot related tonot an element ofa proper subset ofequal tonot equal to the set
B
.

The **Cartesian product** of the sets \( A \) and \( B \), denoted by \( A \times B \), is the set of all **ordered pairs** \( (a, b) \), where \( a \) is **an element of** the set \( A \), and \( b \) is **an element of** the set \( B \).

In mathematical notation:

\[
A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}
\]

This means that the Cartesian product consists of all possible pairs where the first element comes from \( A \) and the second element comes from \( B \).

Would you like further explanation or examples?

Here are 5 related questions to expand on this topic:
1. How would you define the Cartesian product of three sets, \( A \times B \times C \)?
2. What is the Cartesian product of a set \( A \) with itself, i.e., \( A \times A \)?
3. How does the Cartesian product differ from the union of sets?
4. What are some applications of Cartesian products in real-life scenarios or other mathematical fields?
5. If \( A = \{1, 2\} \) and \( B = \{3, 4\} \), what is \( A \times B \)?

**Tip:** The Cartesian product is foundational in creating coordinate systems, where elements from two sets combine to form points in a plane.

Let A and B be sets. The Cartesian product of the sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a is an element of the set A and b is an element of the set B.

Explore the Cartesian product of two sets A and B, including its definition, ordered pairs, and mathematical notation. Learn how to calculate A × B with examples and applications. Ideal for high school students studying set theory.

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