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Question: What is the Cartesian product of A, B, and C, where:

• $A = \{1, 2\}$,
• $B = \{x, y\}$,
• $C = \{0, 2\}$?

The Cartesian product $A \times B \times C$ includes all possible ordered triples formed by selecting one element from $A$, one from $B$, and one from $C$.

Let's solve this step by step:

Solution:

Given:

• $A = \{1, 2\}$,
• $B = \{x, y\}$,
• $C = \{0, 2\}$,

The Cartesian product $A \times B \times C$ is the set of all ordered triples $(a, b, c)$ such that $a \in A$, $b \in B$, and $c \in C$.

We can now compute all combinations:

• From $A = \{1, 2\}$,
• From $B = \{x, y\}$,
• From $C = \{0, 2\}$,

The possible triples are:

1. $(1, x, 0)$,
2. $(1, x, 2)$,
3. $(1, y, 0)$,
4. $(1, y, 2)$,
5. $(2, x, 0)$,
6. $(2, x, 2)$,
7. $(2, y, 0)$,
8. $(2, y, 2)$.

Thus, the Cartesian product is: $A \times B \times C = \{(1, x, 0), (1, x, 2), (1, y, 0), (1, y, 2), (2, x, 0), (2, x, 2), (2, y, 0), (2, y, 2)\}$

Would you like further details or have any questions on this?

Related questions:

1. What is the Cartesian product of two sets, for example, $A \times B$?
2. How many elements are in the Cartesian product of three sets?
3. Can the Cartesian product be defined for infinite sets?
4. What is the relation between Cartesian product and relations in set theory?
5. How is the Cartesian product useful in database management and SQL?

Tip:

The size of the Cartesian product of sets $A$, $B$, and $C$ is the product of their cardinalities. For instance, if $|A| = 2$, $|B| = 2$, and $|C| = 2$, then $|A \times B \times C| = 2 \times 2 \times 2 = 8$.