If `[.]` denotes the greatest integer function and `x,yepsilonRr,"n" epsilonN` then which of the following is true? (A) (B) (C) (D)

Options

A. a. `[x+y]ge[x]+[y]`
B. b. `[x+y]le[x]+[y]`
C. c. `[([x])/n]=[x/n]`
D. d. `[x+1/2]=[2x]-[x]`

Explanation

To solve the problem, we need to analyze the given options involving the greatest integer function (denoted as `[.]`). The greatest integer function, also known as the floor function, returns the largest integer less than or equal to a given number.

Let's evaluate each option step by step.

### Step 1: Analyze Option A
**Statement**: \([x + y] \geq [x] + [y]\)

**Proof**:
1. Let \(x = i_1 + f_1\) and \(y = i_2 + f_2\), where \(i_1\) and \(i_2\) are integers, and \(f_1\) and \(f_2\) are the fractional parts such that \(0 \leq f_1, f_2 < 1\).
2. Then, \(x + y = (i_1 + i_2) + (f_1 + f_2)\).
3. The greatest integer of \(x + y\) can be expressed as:
\[ [x + y] = [i_1 + i_2 + f_1 + f_2] \]
- If \(f_1 + f_2 < 1\), then \([x + y] = i_1 + i_2\).
- If \(f_1 + f_2 \geq 1\), then \([x + y] = i_1 + i_2 + 1\).
4. On the right-hand side:
\[ [x] + [y] = [i_1 + f_1] + [i_2 + f_2] = i_1 + i_2 \]
5. Therefore, we can conclude:
\[ [x + y] \geq [x] + [y] \]
This confirms that Option A is true.

### Step 2: Analyze Option B
**Statement**: (Not provided)

### Step 3: Analyze Option C
**Statement**: \([x/n] = [x]/n\) for \(n \in \mathbb{N}\)

**Proof**:
1. Let \(x = i + f\) where \(i\) is an integer and \(f\) is the fractional part.
2. Then:
\[ [x/n] = [i/n + f/n] \]
- If \(i/n\) is an integer, then \([x/n] = i/n\).
- If \(i/n\) is not an integer, then \([x/n] = i/n\) or \((i/n) + 1\) depending on \(f/n\).
3. On the right-hand side:
\[ [x]/n = i/n \]
4. Therefore, since the left-hand side can vary based on the fractional part, this option is not necessarily true.

### Step 4: Analyze Option D
**Statement**: \([x + 1/2] = [2x] - [x]\)

**Proof**:
1. Let \(x = i + f\) where \(i\) is an integer and \(f\) is the fractional part.
2. Then:
\[ [x + 1/2] = [i + f + 1/2] \]
- If \(f < 1/2\), then \([x + 1/2] = i\).
- If \(f \geq 1/2\), then \([x + 1/2] = i + 1\).
3. On the right-hand side:
\[ [2x] = [2(i + f)] = [2i + 2f] \]
- If \(2f < 1\), then \([2x] = 2i\).
- If \(1 \leq 2f < 2\), then \([2x] = 2i + 1\).
4. Thus, we can see that both sides can be equal under the conditions analyzed, confirming that Option D is true.

### Conclusion
From the analysis, we conclude that:
- Option A is true.
- Option C is false.
- Option D is true.