If `bar(z)` be the conjugate of the complex number z, then which of the following relations is false ?

Options

A. `|Z| = | bar(Z) |`
B. `Z. bar(Z) = | bar(Z)|^(2)`
C. `bar( Z_(1) +Z_(2)) = bar( Z)_(1) + bar(Z)_(2)`
D. arg `(Z) cancel(=) arg (bar(z))`

Answer

To solve the problem of identifying which relation involving the conjugate of a complex number \( z \) is false, we need to analyze the properties of complex conjugates.

Let's denote the complex number \( z \) as \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

The conjugate of \( z \), denoted as \( \bar{z} \), is given by:
\[ \bar{z} = a - bi \]

Now, we will evaluate common properties of complex conjugates:

1. **Conjugate of a Sum**:
\[ \overline{(z_1 + z_2)} = \bar{z_1} + \bar{z_2} \]
This property states that the conjugate of the sum of two complex numbers is equal to the sum of their conjugates.

2. **Conjugate of a Product**:
\[ \overline{(z_1 \cdot z_2)} = \bar{z_1} \cdot \bar{z_2} \]
This property states that the conjugate of the product of two complex numbers is equal to the product of their conjugates.

3. **Conjugate of a Conjugate**:
\[ \overline{\bar{z}} = z \]
This property states that the conjugate of the conjugate of a complex number returns the original complex number.

4. **Conjugate of a Real Number**:
\[ \overline{a} = a \quad \text{(for any real number } a\text{)} \]
This property states that the conjugate of a real number is the number itself.

5. **Conjugate of a Scalar Multiple**:
\[ \overline{(c \cdot z)} = c \cdot \bar{z} \quad \text{(for any real number } c\text{)} \]
This property states that the conjugate of a scalar multiple of a complex number is equal to the scalar multiplied by the conjugate of the complex number.

Now, to identify which relation is false, we need to consider the options provided (which were not included in the question). However, based on the properties listed, we can conclude that any statement contradicting these properties would be false.

### Steps Recap:
1. Define the complex number \( z \) and its conjugate \( \bar{z} \).
2. List the properties of complex conjugates.
3. Identify which of the properties is contradicted by the options given.