Let's examine each statement individually to determine which ones are true.

1. Statement: $(z_1 + z_2) z_3 = z_1 z_3 + z_2 z_3$

   • This statement checks if the distributive property holds for complex numbers under multiplication and addition. This property is indeed valid for complex numbers, as they follow the distributive property just like real numbers.
   • Conclusion: True.

2. Statement: "The set of complex numbers with the operation of multiplication is a commutative semigroup."

   • A commutative semigroup is a set that is closed under an associative and commutative operation, but does not necessarily have an identity or inverse elements.
   • The set of complex numbers under multiplication is indeed closed, associative, and commutative, so it qualifies as a commutative semigroup.
   • Conclusion: True.

3. Statement: "The neutral element in the set of complex numbers with the operation of addition is $z = 1$."

   • In the context of addition, the neutral (or identity) element for complex numbers is $z = 0$, not $z = 1$.
   • Conclusion: False.

4. Statement: $|z_1 z_2| = |z_1| \cdot |z_2|$

   • This property refers to the modulus of a product of two complex numbers. It is a known property that the modulus of the product of two complex numbers equals the product of their moduli.
   • Conclusion: True.

Summary:

The correct statements are:

• Statement 1: $(z_1 + z_2) z_3 = z_1 z_3 + z_2 z_3$
• Statement 2: "The set of complex numbers with the operation of multiplication is a commutative semigroup."
• Statement 4: $|z_1 z_2| = |z_1| \cdot |z_2|$

Would you like more details on any of these?

Here are five related questions for further exploration:

1. What is the identity element for multiplication in the set of complex numbers?
2. Can complex numbers form a group under multiplication?
3. How does the modulus operation behave under addition of complex numbers?
4. Is the set of complex numbers closed under division?
5. What are the properties of complex conjugation with respect to addition and multiplication?

Tip: When working with complex numbers, keep in mind their properties under both addition and multiplication, as these operations have distinct neutral elements and behaviors.