The Last Step in a Proof Contains the Conclusion
In mathematics, a proof is a series of logical steps that demonstrate the truth of a theorem or proposition. The last step in a proof is the conclusion, which is the statement that is being proven.
The conclusion of a proof is always the statement that is being proven. This is because the purpose of a proof is to show that the conclusion is true. The other steps in the proof are used to build up to the conclusion, using logical reasoning.
For example, consider the following proof of the Pythagorean Theorem:
Given: * Triangle ABC is a right triangle, with right angle at C. * AC = a * BC = b To prove: * AB^2 = a^2 + b^2 Proof: 1. By the Pythagorean Theorem, * AC^2 = AB^2 + BC^2 2. Substituting a for AC and b for BC, * a^2 = AB^2 + b^2 3. Therefore, * AB^2 = a^2 + b^2 The conclusion of this proof is the statement "AB^2 = a^2 + b^2". This is the statement that is being proven, and it is the last step in the proof.
Here are some questions that you can ask yourself to help you identify the conclusion of a proof:
- What is the statement that is being proven?
- What is the last step in the proof?
- Is the last step in the proof a logical conclusion from the previous steps?
If you can answer these questions, then you will be able to identify the conclusion of a proof.
Here are some additional thoughts on the last step in a proof:
- The conclusion of a proof should be clear and concise. It should be easy to understand what the conclusion is.
- The conclusion of a proof should be supported by the previous steps. The previous steps should logically lead to the conclusion.
- The conclusion of a proof should be true. If the conclusion is not true, then the proof is flawed.
