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- A “two-column proof” is a tool that we use to learn techniques for writing proofs.
- The left-hand column contains a sequence of assertions.
- The right-hand column contains a justification for each assertion.
- Each row of the proof is numbered (in the left margin) for easy reference.
- A dark horizontal line is drawn to indicate the end of the hypotheses.
- A dark horizontal line is drawn along the left edge of the proof, and of each subproof.
- In addition to the basic theorems of , we have two rules that use subproofs:
- proof by contradiction
- Proofs often use the Law of Excluded Middle, the Rules of Negation, and contrapositives.
- Assertions that are in a subproof cannot be used as justification for lines that are not in that same subproof.
- Writing proofs takes practice, but there are some strategies that can help.
- Proofs can also be written in English prose, using sentences and paragraphs.
- To show that a deduction is not valid, find a counterexample.