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Unformatted text preview: CPSC 121 Lecture 7 January 19, 2009 Menu January 19, 2009 Topics: Proofs (in Propositional Logic) Rules of Inference Reading: Today: Epp 1.3 Next: Epp 2.1, 2.3 Reminders: On-line Quiz 5 deadline 9:00pm January 22 Assignment 1 due Friday, January 30, 17:00 READ the WebCT Vista course announcements board WebCT Vista: http://www.vista.ubc.ca www: http://www.ugrad.cs.ubc.ca/~cs121/ Proofs (in Propositional Logic) What is a Proof? A proof is a method to determine “truth.” There are many approaches... 1. Justice System: Proof “beyond a reasonable doubt” (criminal) or on the “preponderance of the evidence” (civil) as determined by a jury of 12 people selected at random 2. Religion: Truth revealed by “God,” perhaps via a third party 3. Authority: Truth obtained from someone with whom it is unwise to disagree 4. Experimental Science: Truth is guessed and hypotheses are confirmed or refuted by experiment 5. Sampling: Truth by statistical analysis. For example, public opinion determined by polling a representative sample 6. Democracy: Truth obtained by vote Mathematical Proof Mathematics has a specific notion of “proof” Definition: A mathematical proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms Definition: An axiom is a proposition that is assumed to be true Definition. A set of axioms is consistent if no proposition can be proven to be both true and false Definition. A set of axioms is complete if it can be used to prove or disprove every proposition Definition: Logical deductions (aka inference rules ) are used to combine axioms and true propositions to construct more true propositions. Proofs are important. They permit you to convince yourself and others that your reasoning is correct. They help you to understand why something is true and whether it will stay true when other things change. In order to be certain everyone is working with the same knowledge, let’s review some terminology. Terminology Definition: A theorem is a proposition that has been proven to be true ASIDE: A proposition remains a conjecture if not yet proven to be true (and not yet known to be false) Definition: A lemma is a theorem that may not be too interesting in its own right but that is useful in proving another theorem Definition: A...
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This note was uploaded on 01/09/2010 for the course CPSC 121 taught by Professor Belleville during the Spring '08 term at UBC.
- Spring '08