If n(A) = 3 and n(B) = 4, then what is n(A x B)?
Understand the Problem
The question is asking about the cardinality of the Cartesian product of two sets. Given the number of elements in set A (n(A) = 3) and the number of elements in set B (n(B) = 4), we need to find the number of elements in the set A x B, which is the Cartesian product of A and B.
Answer
b) 12
Answer for screen readers
b) 12
Steps to Solve
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Understand the Cartesian Product The Cartesian product of two sets A and B, denoted as $A \times B$, is the set of all ordered pairs (a, b) where a is in A and b is in B.
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Formula for Cardinality of Cartesian Product The cardinality (number of elements) of the Cartesian product $A \times B$ is given by the product of the cardinalities of A and B. That is: $$n(A \times B) = n(A) \times n(B)$$
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Apply the Formula Given $n(A) = 3$ and $n(B) = 4$, we can calculate $n(A \times B)$ as follows: $$n(A \times B) = 3 \times 4 = 12$$
b) 12
More Information
The Cartesian product is fundamental in set theory and is used in various areas of mathematics, including relations, functions, and combinatorics. The cardinality of the Cartesian product is a simple but important concept.
Tips
A common mistake is to add the number of elements instead of multiplying them. Remember, the number of elements in $A \times B$ is the product of the number of elements in A and B, not the sum. Another possible mistake is to confuse the concept of Cartesian product with other set operations like union or intersection.
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