2.8: Summary

Summary

• 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:
  • \(\Rightarrow\)-introduction
  • 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.