a and b are 2 sets such that n(a)=2  n(b)=4 and n(a union b)=4 then find n((a cross a)intersection (b cross b))

We are given two sets \( A \) and \( B \) with the following information:

- \( n(A) = 2 \) (i.e., set \( A \) has 2 elements)
- \( n(B) = 4 \) (i.e., set \( B \) has 4 elements)
- \( n(A \cup B) = 4 \) (the union of \( A \) and \( B \) contains 4 elements)

We are asked to find \( n((A \times A) \cap (B \times B)) \).

### Step 1: Understand the relation between \( A \) and \( B \)
Since \( n(A \cup B) = 4 \), this means that all elements of \( A \) must be present in \( B \) because \( A \) has 2 elements and the union has 4 elements. Thus, \( A \subseteq B \).

So, \( A \) contains 2 elements, and \( B \) contains 4 elements, with all of \( A \)'s elements being in \( B \).

### Step 2: Cartesian products
We need to compute \( A \times A \) and \( B \times B \):
- \( A \times A \) is the set of ordered pairs where both elements are from \( A \). Since \( A \) has 2 elements, \( n(A \times A) = 2 \times 2 = 4 \) pairs.
- \( B \times B \) is the set of ordered pairs where both elements are from \( B \). Since \( B \) has 4 elements, \( n(B \times B) = 4 \times 4 = 16 \) pairs.

### Step 3: Intersection of \( A \times A \) and \( B \times B \)
The intersection \( (A \times A) \cap (B \times B) \) contains only those ordered pairs that are in both \( A \times A \) and \( B \times B \). Since \( A \subseteq B \), all pairs from \( A \times A \) will also be in \( B \times B \).

Thus, \( n((A \times A) \cap (B \times B)) = n(A \times A) = 4 \).

### Final Answer:
The number of elements in \( (A \times A) \cap (B \times B) \) is \( \boxed{4} \).

Would you like more details or have any other questions? Here are some related questions for practice:

1. If \( n(A) = 3 \) and \( n(B) = 5 \), what is \( n(A \times A) \)?
2. If \( A \) and \( B \) are disjoint, what is \( n(A \times A) \cap (B \times B) \)?
3. If \( n(A) = 2 \), \( n(B) = 4 \), and \( n(A \cap B) = 1 \), what is \( n(A \times A) \cap (B \times B) \)?
4. Find \( n(A \cup B) \) if \( A \subseteq B \), \( n(A) = 3 \), and \( n(B) = 5 \).
5. Calculate \( n((A \times B) \cup (B \times A)) \) given \( n(A) = 3 \) and \( n(B) = 4 \).

**Tip**: When dealing with Cartesian products, remember that the size of the product \( A \times A \) is \( n(A)^2 \), and intersections often depend on how sets relate to each other.

a and b are 2 sets such that n(a)=2, n(b)=4, and n(a union b)=4. Then find n((a cross a) intersection (b cross b))

This problem involves finding the number of elements in the intersection of Cartesian products of two sets, A and B. Given n(A) = 2, n(B) = 4, and n(A ∪ B) = 4, the solution shows how to calculate n((A × A) ∩ (B × B)). Key concepts include set theory and Cartesian products. Suitable for Grades 10-12.

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