[m] is defined as the greater integer less than m.
{m} is defined as the least integer greater than m.
f(x,y) = {x}+[y]
g(x,y) = [x] - {y}
F(f(x,y)) = {f(x,y)} - [g(x,y)]
G(g(x,y)) = [f(x,y)] - {g(x,y)}
If x and y are consecutive integers, then find g(x,y):

Options

A. -1
B. -3
C. `-1 or -3`
D. none of these

Answer

To solve the problem, we need to find \( g(x, y) \) given that \( x \) and \( y \) are consecutive integers. Let's denote the consecutive integers as follows:

Let \( x = n \) (where \( n \) is an integer) and \( y = n + 1 \).

Now, we will compute \( g(x, y) \) using the formula provided:

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

### Step 1: Calculate \([x]\)
Since \( x = n \), we have:
\[ [x] = [n] \]
By definition, \([n]\) is the greatest integer less than or equal to \( n \), which is simply \( n \) itself.

### Step 2: Calculate \(\{y\}\)
Now, we calculate \(\{y\}\):
\[ \{y\} = \{n + 1\} \]
By definition, \(\{n + 1\}\) is the least integer greater than \( n + 1 \), which is \( n + 2 \).

### Step 3: Substitute into \( g(x, y) \)
Now we substitute the values we found into the formula for \( g(x, y) \):
\[ g(x, y) = [x] - \{y\} = n - (n + 2) \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ g(x, y) = n - n - 2 = -2 \]

### Final Answer
Thus, the value of \( g(x, y) \) when \( x \) and \( y \) are consecutive integers is:
\[ \boxed{-2} \]