Some elements of `AxxB` are `(a,x),(c,y),(d,z)`. If `A={a,b,c,d}`, find the remaining elements of `AxxB` such that `n(AxxB)` is least.

Answer

To solve the problem, we need to find the remaining elements of the Cartesian product \( A \times B \) given that some elements are already known.

### Step-by-Step Solution:

1. **Identify the Sets:**
- We have \( A = \{a, b, c, d\} \).
- The elements of \( A \times B \) given are \( (a,x), (c,y), (d,z) \).

2. **Determine Elements of Set B:**
- From the ordered pairs, we can deduce that:
- The element \( x \) corresponds to \( a \).
- The element \( y \) corresponds to \( c \).
- The element \( z \) corresponds to \( d \).
- Therefore, we can conclude that the elements of set \( B \) must include at least \( \{x, y, z\} \).

3. **Minimize the Number of Elements in B:**
- To minimize \( n(A \times B) \), we should keep the number of elements in \( B \) as small as possible while still including \( x, y, z \).
- Thus, we can take \( B = \{x, y, z\} \).

4. **Calculate the Total Elements in \( A \times B \):**
- The Cartesian product \( A \times B \) will have \( n(A) \times n(B) \) elements.
- Here, \( n(A) = 4 \) (since \( A \) has 4 elements) and \( n(B) = 3 \) (since \( B \) has 3 elements).
- Therefore, \( n(A \times B) = 4 \times 3 = 12 \).

5. **List All Elements of \( A \times B \):**
- The elements of \( A \times B \) will be:
- For \( a \): \( (a,x), (a,y), (a,z) \)
- For \( b \): \( (b,x), (b,y), (b,z) \)
- For \( c \): \( (c,x), (c,y), (c,z) \)
- For \( d \): \( (d,x), (d,y), (d,z) \)

6. **Identify Remaining Elements:**
- From the total \( 12 \) elements in \( A \times B \), we already have \( (a,x), (c,y), (d,z) \).
- The remaining elements are:
- From \( a \): \( (a,y), (a,z) \)
- From \( b \): \( (b,x), (b,y), (b,z) \)
- From \( c \): \( (c,x), (c,z) \)
- From \( d \): \( (d,x), (d,y) \)

### Remaining Ordered Pairs:
- The remaining ordered pairs in \( A \times B \) are:
- \( (a,y), (a,z) \)
- \( (b,x), (b,y), (b,z) \)
- \( (c,x), (c,z) \)
- \( (d,x), (d,y) \)

### Final Answer:
The remaining elements of \( A \times B \) are:
- \( (a,y), (a,z), (b,x), (b,y), (b,z), (c,x), (c,z), (d,x), (d,y) \).