Given `A={2,3,4,5,6}`. List the elements of each of the following relations:
`{(x,y)inAxxA:xgty,(x)/(y)cancelinW}`

Answer

To solve the problem, we need to find the elements of the relation defined by the conditions:

1. \( x > y \)
2. \( \frac{x}{y} \notin W \) (where \( W \) is the set of whole numbers)

Given the set \( A = \{2, 3, 4, 5, 6\} \), we will follow these steps:

### Step 1: Identify the Cartesian Product \( A \times A \)

The Cartesian product \( A \times A \) consists of all ordered pairs \( (x, y) \) where both \( x \) and \( y \) are elements of \( A \).

\[ A \times A = \{(2,2), (2,3), (2,4), (2,5), (2,6), (3,2), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6)\} \]

### Step 2: Apply the Condition \( x > y \)

Now, we filter the pairs to satisfy the condition \( x > y \):

- For \( x = 2 \): No valid \( y \) (since \( 2 \) is the smallest).
- For \( x = 3 \): Valid \( y \) is \( 2 \) → Pair: \( (3, 2) \)
- For \( x = 4 \): Valid \( y \) are \( 2, 3 \) → Pairs: \( (4, 2), (4, 3) \)
- For \( x = 5 \): Valid \( y \) are \( 2, 3, 4 \) → Pairs: \( (5, 2), (5, 3), (5, 4) \)
- For \( x = 6 \): Valid \( y \) are \( 2, 3, 4, 5 \) → Pairs: \( (6, 2), (6, 3), (6, 4), (6, 5) \)

So, the pairs satisfying \( x > y \) are:

\[ \{(3, 2), (4, 2), (4, 3), (5, 2), (5, 3), (5, 4), (6, 2), (6, 3), (6, 4), (6, 5)\} \]

### Step 3: Apply the Condition \( \frac{x}{y} \notin W \)

Next, we check which of these pairs do not yield whole numbers when \( x \) is divided by \( y \):

- \( (3, 2) \): \( \frac{3}{2} = 1.5 \) (valid)
- \( (4, 2) \): \( \frac{4}{2} = 2 \) (not valid)
- \( (4, 3) \): \( \frac{4}{3} \approx 1.33 \) (valid)
- \( (5, 2) \): \( \frac{5}{2} = 2.5 \) (valid)
- \( (5, 3) \): \( \frac{5}{3} \approx 1.67 \) (valid)
- \( (5, 4) \): \( \frac{5}{4} = 1.25 \) (valid)
- \( (6, 2) \): \( \frac{6}{2} = 3 \) (not valid)
- \( (6, 3) \): \( \frac{6}{3} = 2 \) (not valid)
- \( (6, 4) \): \( \frac{6}{4} = 1.5 \) (valid)
- \( (6, 5) \): \( \frac{6}{5} = 1.2 \) (valid)

After applying the second condition, the valid pairs are:

\[ \{(3, 2), (4, 3), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5)\} \]

### Final Result

The elements of the relation are:

\[ \{(3, 2), (4, 3), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5)\} \]