To solve this problem, we need to find the Cartesian products $A \times B$ and $B \times A$, and then determine if they are equal.

Step 1: Find $A \times B$

The Cartesian product $A \times B$ consists of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Given: $A = \{0, 2\} \quad \text{and} \quad B = \{4, 5, 6\}$

The elements of $A \times B$ are:

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

Step 2: Find $B \times A$

The Cartesian product $B \times A$ consists of all ordered pairs $(b, a)$ where $b \in B$ and $a \in A$.

The elements of $B \times A$ are:

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

Step 3: Check if $A \times B = B \times A$

For two sets to be equal, all elements in one set must be present in the other, and the order of pairs matters in Cartesian products.

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

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

Clearly, the ordered pairs are not the same because the first and second elements in each pair are reversed.

Conclusion:

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

Thus, the relation $A \times B = B \times A$ does not hold.

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Would you like to explore more details, or do you have any questions?

Related Questions:

1. How is the Cartesian product defined for sets with more than two elements?
2. Can the Cartesian product be defined for three or more sets?
3. What are some practical applications of Cartesian products in mathematics?
4. How does the order of elements affect the Cartesian product?
5. What is the significance of ordered pairs in Cartesian products?

Tip: Remember, in Cartesian products, the order of the elements in the pairs is crucial and changes the identity of the product.