Simplify `x - [ y - { z - (2x - bar(y+z)) }] `

Answer

To simplify the expression \( x - [ y - \{ z - (2x - \overline{y+z}) \} ] \), we will follow these steps:

### Step-by-Step Solution:

1. **Write Down the Expression**:
\[ x - [ y - \{ z - (2x - \overline{y+z}) \} ] \]

2. **Simplify the Overline**:
The overline indicates negation. Thus, we can express \( \overline{y+z} \) as \( -y - z \). Therefore, the expression becomes:
\[ x - [ y - \{ z - (2x + y + z) \} ] \]

3. **Simplify Inside the Curly Bracket**:
Now, simplify the expression inside the curly brackets:
\[ z - (2x + y + z) = z - 2x - y - z = -2x - y \]
So, we have:
\[ x - [ y - (-2x - y) ] \]

4. **Simplify the Bracket**:
Now, simplify the expression inside the square brackets:
\[ y - (-2x - y) = y + 2x + y = 2y + 2x \]
Thus, the expression now looks like:
\[ x - (2y + 2x) \]

5. **Distribute the Negative Sign**:
Now distribute the negative sign:
\[ x - 2y - 2x \]

6. **Combine Like Terms**:
Combine the \( x \) terms:
\[ (1x - 2x) - 2y = -x - 2y \]

7. **Final Result**:
The simplified expression is:
\[ -x - 2y \]