If [x] read as the greatest integer less than or equal to `x, {x}` is the least integer greater than or equal to x, Further, `f(x, y) = [x] + {y}` and `g(x, y) = {x} - {y}`
and `P (x,y) = f(x , y) +g(x,y)`
and `Q (x,y) = f(x,y) -g(x,y)`
If `x, y in I^(+) ` then `P(x, y) + Q(x,y)` is always:

Options

A. A) an even number
B. B) an odd number
C. C) can't say
D. D) none of these

Answer

To solve the problem, we need to analyze the functions \( f(x, y) \), \( g(x, y) \), \( P(x, y) \), and \( Q(x, y) \) step by step.

### Step 1: Understand the functions

1. **Greatest Integer Function**: \([x]\) is the greatest integer less than or equal to \(x\).
2. **Least Integer Function**: \({x}\) is the least integer greater than or equal to \(x\).

### Step 2: Define the functions

- \( f(x, y) = [x] + {y} \)
- \( g(x, y) = {x} - {y} \)

### Step 3: Define \( P(x, y) \) and \( Q(x, y) \)

- \( P(x, y) = f(x, y) + g(x, y) \)
- \( Q(x, y) = f(x, y) - g(x, y) \)

### Step 4: Calculate \( P(x, y) \)

Substituting the definitions of \( f \) and \( g \):

\[ P(x, y) = ([x] + {y}) + ({x} - {y}) \]
\[ P(x, y) = [x] + {y} + {x} - {y} \]
\[ P(x, y) = [x] + {x} \]

### Step 5: Calculate \( Q(x, y) \)

Substituting the definitions of \( f \) and \( g \):

\[ Q(x, y) = ([x] + {y}) - ({x} - {y}) \]
\[ Q(x, y) = [x] + {y} - {x} + {y} \]
\[ Q(x, y) = [x] - {x} + 2{y} \]

### Step 6: Combine \( P(x, y) \) and \( Q(x, y) \)

Now, we need to find \( P(x, y) + Q(x, y) \):

\[ P(x, y) + Q(x, y) = ([x] + {x}) + ([x] - {x} + 2{y}) \]
\[ = [x] + {x} + [x] - {x} + 2{y} \]
\[ = 2[x] + 2{y} \]

### Step 7: Conclusion

Since \( x \) and \( y \) are positive integers, \([x]\) is an integer and \({y}\) is also an integer. Therefore, \( P(x, y) + Q(x, y) = 2([x] + {y}) \) will always be an integer.

### Final Answer

Thus, \( P(x, y) + Q(x, y) \) is always an integer.

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