Given the sets $A = \{0, 2\}$ and $B = \{4, 5, 6\}$, we need to list the elements of the Cartesian products $A \times B$ and $B \times A$.

Elements of $A \times B$

The Cartesian product $A \times B$ is the set of all ordered pairs where the first element is from $A$ and the second element is from $B$. Therefore:

$A \times B = \{(0, 4), (0, 5), (0, 6), (2, 4), (2, 5), (2, 6)\}$

Elements of $B \times A$

Similarly, the Cartesian product $B \times A$ is the set of all ordered pairs where the first element is from $B$ and the second element is from $A$. Therefore:

$B \times A = \{(4, 0), (4, 2), (5, 0), (5, 2), (6, 0), (6, 2)\}$

Comparing $A \times B$ and $B \times A$

To determine if $A \times B = B \times A$, we need to compare the elements of both sets.

• $A \times B$ contains the pairs $(0, 4), (0, 5), (0, 6), (2, 4), (2, 5), (2, 6)$.
• $B \times A$ contains the pairs $(4, 0), (4, 2), (5, 0), (5, 2), (6, 0), (6, 2)$.

Since $A \times B$ and $B \times A$ consist of different ordered pairs (each pair in $A \times B$ has its elements reversed in $B \times A$), we conclude:

$A \times B \neq B \times A$

This shows that the relation $A \times B = B \times A$ does not hold.

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Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How does the Cartesian product change if one of the sets is empty?
2. What happens to the Cartesian product if the sets $A$ and $B$ are identical?
3. Can the Cartesian product of two sets ever be commutative under any circumstances?
4. How does the size of the Cartesian product relate to the sizes of the sets $A$ and $B$?
5. What are some practical applications of Cartesian products in computer science?

Tip: The Cartesian product is not commutative, meaning $A \times B$ generally does not equal $B \times A$. The order of elements in each pair matters.