Deductive Reasoning and Proving
In summary, below are the steps in writing a direct proof.
| 1. | Illustrate what is to be proven by drawing the necessary figure(s). |
| 2. | List the given statements. Write the conclusion that you need to prove. |
| 3. | Mark the figure(s) as needed, based on what you can deduce from the given statements. |
| 4. | Write all statements, including the simplest details, along with the reasons in a two-column table. Oftentimes, the first steps are the given statements and the last step is the conclusion that you need to prove. |
There are cases when an indirect method of proving can be more easily done. The following are the steps in writing an indirect proof:
| 1. | Assume that the conclusion is false. |
| 2. | Show that the assumption leads to a contradiction of the hypothesis or a known statement that is true. |
| 3. | Since the assumption about the conclusion is false, then the conclusion must be true. |
To use the indirect method to prove the implication "if p, then q," assume the hypothesis p and the negation of the conclusion "not q."
Link these assumptions with definitions or other statements that are considered true to establish "if p and not q, then r." However, "not r" is known.
You arrive with a contradiction "r and not r." Therefore, the statement "if p, then q" is true.
