How to Write an Indirect Proof
Start with two possible statements., Write down what you know is true., Assume one of the statements is true., Draw logical conclusions and look for contradictions., Conclude that your assumption was incorrect., Infer that the other statement must...
Step-by-Step Guide
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Step 1: Start with two possible statements.
Indirect proofs work if you can describe the situation in two possible ways.
Since there are only two options, once you prove one statement wrong, you will know the other one is correct.
These are usually just two opposites: "A is true" and "A is not true." 'Example:' Think of a suspect in a police investigation.
There are two possible explanations: the suspect is innocent; or the suspect is guilty.
If we can rule out the idea that he's guilty, we automatically know he is innocent. -
Step 2: Write down what you know is true.
These statements are often called "axioms" or "givens" (as in, the information given to you).
You don't have to write down every fact you know, but it may help to write down related, proven statements.
These can help you draw logical conclusions.
Example: "The person who committed the crime was at the crime scene." and "A person cannot be in two places at once." are two examples of real-life "givens." These should be so obvious you can include them in your proof without needing evidence. , Pick the one you think you can disprove most easily.
Start with the idea "what if this statement is actually true?" This is called a postulate.
The goal of the indirect proof is to show where this postulate leads.
Example:
Assume the suspect is guilty.
It might not be true, but that's what this proof will tell us. , Why is it useful to assume something that might not be true? The goal is not to find out the truth, but to look for contradictions.
If your assumption leads to two contradictory statements, or if it contradicts one of your "givens," it means your assumption must be wrong.
Example:
If the suspect is guilty, as you've assumed, he must have been present while the crime was committed.
Witnesses saw the suspect in a different city on the day of the crime.
These two facts contradict each other.
If you fail to find any contradictions, it does not mean your assumption was correct, only that it is possible. , If you found a contradiction, and there are no faults with your logic, your initial assumption must have been wrong.
Example:
The suspect could not have committed the crime and been in a different city at the same time.
Therefore, the assumption that the suspect was guilty must be incorrect. , Now you know one statement is incorrect.
Since there is only one other possible statement, that one must be right.
You have now proved this statement indirectly.
Example:
Since the suspect cannot be guilty, he must be innocent.
Notice that you do not need to spend any time investigating the other statement. -
Step 3: Assume one of the statements is true.
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Step 4: Draw logical conclusions and look for contradictions.
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Step 5: Conclude that your assumption was incorrect.
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Step 6: Infer that the other statement must be correct.
Detailed Guide
Indirect proofs work if you can describe the situation in two possible ways.
Since there are only two options, once you prove one statement wrong, you will know the other one is correct.
These are usually just two opposites: "A is true" and "A is not true." 'Example:' Think of a suspect in a police investigation.
There are two possible explanations: the suspect is innocent; or the suspect is guilty.
If we can rule out the idea that he's guilty, we automatically know he is innocent.
These statements are often called "axioms" or "givens" (as in, the information given to you).
You don't have to write down every fact you know, but it may help to write down related, proven statements.
These can help you draw logical conclusions.
Example: "The person who committed the crime was at the crime scene." and "A person cannot be in two places at once." are two examples of real-life "givens." These should be so obvious you can include them in your proof without needing evidence. , Pick the one you think you can disprove most easily.
Start with the idea "what if this statement is actually true?" This is called a postulate.
The goal of the indirect proof is to show where this postulate leads.
Example:
Assume the suspect is guilty.
It might not be true, but that's what this proof will tell us. , Why is it useful to assume something that might not be true? The goal is not to find out the truth, but to look for contradictions.
If your assumption leads to two contradictory statements, or if it contradicts one of your "givens," it means your assumption must be wrong.
Example:
If the suspect is guilty, as you've assumed, he must have been present while the crime was committed.
Witnesses saw the suspect in a different city on the day of the crime.
These two facts contradict each other.
If you fail to find any contradictions, it does not mean your assumption was correct, only that it is possible. , If you found a contradiction, and there are no faults with your logic, your initial assumption must have been wrong.
Example:
The suspect could not have committed the crime and been in a different city at the same time.
Therefore, the assumption that the suspect was guilty must be incorrect. , Now you know one statement is incorrect.
Since there is only one other possible statement, that one must be right.
You have now proved this statement indirectly.
Example:
Since the suspect cannot be guilty, he must be innocent.
Notice that you do not need to spend any time investigating the other statement.
About the Author
Betty Castillo
Creates helpful guides on cooking to inspire and educate readers.
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