State the axiom or theorem that justifies each step in the following proofs and fill in the blanks. Prove: for all real numbers b and c, [b + (-c)] + c = b.                                                                                                             1. b and c are real numbers  2. -c is a real number  3. [b + (-c)] + c = b + [(-c) + c] 4. ___________ = b + 0  5. _______________ = [-(-a) + (-a)] + a  6. ________________ = 0 + a   7. ___________ = a   8. -(-a) = a

Let's go through the proof step by step, identifying the axioms or theorems that justify each step:

1. **b and c are real numbers.**  
   **Justification:** This is given by the problem statement.

2. **-c is a real number.**  
   **Justification:** **Closure property of real numbers under addition.**  
   (Since c is a real number, its additive inverse, -c, is also a real number.)

3. **[b + (-c)] + c = b + [(-c) + c].**  
   **Justification:** **Associative property of addition.**  
   (The grouping of the numbers in addition can be changed.)

4. **b + [(-c) + c] = b + 0.**  
   **Justification:** **Additive inverse property.**  
   (The sum of a number and its additive inverse is 0, i.e., \((-c) + c = 0\).)

5. **b + 0 = b.**  
   **Justification:** **Identity property of addition.**  
   (Adding 0 to any number gives the number itself.)

6. **[-(-a) + (-a)] + a = -(-a) + [(-a) + a].**  
   **Justification:** **Associative property of addition.**

7. **-(-a) + [(-a) + a] = -(-a) + 0.**  
   **Justification:** **Additive inverse property.**  
   (The sum of a number and its additive inverse is 0, i.e., \((-a) + a = 0\).)

8. **-(-a) + 0 = -(-a).**  
   **Justification:** **Identity property of addition.**  
   (Adding 0 to any number gives the number itself.)

9. **-(-a) = a.**  
   **Justification:** **Double negative property.**  
   (The additive inverse of the additive inverse of a number is the number itself.)

Would you like more details on any specific step or concept?

Here are 5 related questions to expand your understanding:

1. What is the associative property of addition?
2. How does the additive inverse property work for negative numbers?
3. Why is the identity property of addition important in proofs?
4. Can the closure property be applied to multiplication in the same way?
5. What is the difference between an axiom and a theorem?

**Tip:** When working on proofs, always ensure each step is justified by a specific property, axiom, or previously proven theorem. This helps maintain the logical flow and validity of the proof.

Understand and prove the equation [b + (-c)] + c = b for all real numbers b and c. Explore the properties of real numbers and addition in this detailed mathematical proof.

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