Key Concepts and Formulas

• Cardinality of Cartesian Product: For sets $A$ and $B$, $|A \times B| = |A| \times |B|$.
• Total Number of Subsets: A set with $n$ elements has $2^n$ subsets.
• Combinations: The number of ways to choose $k$ elements from a set of $n$ elements is given by $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
• Complementary Counting: The number of desired outcomes = Total outcomes - Number of undesired outcomes.

Step-by-Step Solution

Step 1: Determine the cardinality of A x B

• What we do: Calculate the number of elements in the Cartesian product $A \times B$.
• Why we do it: Knowing the size of $A \times B$ is essential because it tells us the total number of elements we are forming subsets from.
• Given $|A| = 2$ and $|B| = 4$, we have: $|A \times B| = |A| \times |B| = 2 \times 4 = 8$
• Therefore, $A \times B$ has 8 elements. Let $n = |A \times B| = 8$.

Step 2: Define the desired condition

• What we do: Clarify the condition "3 or more elements" in the context of subsets.
• Why we do it: To avoid ambiguity and ensure we're counting the correct subsets.
• The condition means we are interested in subsets of $A \times B$ that have 3, 4, 5, 6, 7, or 8 elements.

Step 3: Choose a counting strategy

• What we do: Decide between direct counting and complementary counting.
• Why we do it: To find the most efficient method. Direct counting would involve calculating $\binom{8}{3} + \binom{8}{4} + \binom{8}{5} + \binom{8}{6} + \binom{8}{7} + \binom{8}{8}$. Complementary counting is more efficient because it only requires calculating the number of subsets with 0, 1, or 2 elements and subtracting from the total number of subsets.
• We will use complementary counting.

Step 4: Calculate the total number of subsets of A x B

• What we do: Calculate the total number of possible subsets of $A \times B$.
• Why we do it: This gives us the starting point for complementary counting.
• Since $|A \times B| = 8$, the total number of subsets is: $2^8 = 256$

Step 5: Calculate the number of "undesired" subsets (0, 1, or 2 elements)

• What we do: Calculate the number of subsets with 0, 1, and 2 elements.
• Why we do it: These are the subsets that do not meet the condition of having 3 or more elements.
  • Number of subsets with 0 elements: $\binom{8}{0} = 1$
  • Number of subsets with 1 element: $\binom{8}{1} = \frac{8!}{1!7!} = 8$
  • Number of subsets with 2 elements: $\binom{8}{2} = \frac{8!}{2!6!} = \frac{8 \times 7}{2 \times 1} = 28$
• The total number of undesired subsets is: $1 + 8 + 28 = 37$

Step 6: Calculate the number of subsets with 3 or more elements

• What we do: Subtract the number of undesired subsets from the total number of subsets.
• Why we do it: This is the final step in complementary counting.
• Number of subsets with 3 or more elements = Total subsets - Undesired subsets $256 - 37 = 219$

Common Mistakes & Tips

• Carefully read "at least" or "at most" conditions. "3 or more" means greater than or equal to 3.
• Always calculate the cardinality of the set you are forming subsets from first.
• Remember to consider the empty set (subset with 0 elements) and the set itself as subsets.

Summary

We used complementary counting to determine the number of subsets of $A \times B$ (where $|A \times B| = 8$) that have 3 or more elements. We calculated the total number of subsets ($2^8 = 256$) and subtracted the number of subsets with 0, 1, or 2 elements (37) to obtain the answer, 219.

The final answer is \boxed{219}, which corresponds to option (A).