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Logic (Computer Science Notes)
Proposition
•
A proposition is a statement which is true or false (but never both!). For example,
“Urbana is in Illinois” or 2 < 15. It can’t be a question. It also can’t contain variables, e.g.
x < 9 isn’t a proposition. Sentence fragments without verbs (e.g. “bright blue flowers”) or
arithmetic expressions (e.g. 5 + 17), aren’t propositions because they don’t state a claim.
Implication
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Two propositions p and q can also be joined into the conditional statement. “if p, then q.”
The proposition after the “if” (p in this case) is called the “hypothesis” and the
proposition after “then” (q in this example) is called the “conclusion.” As in normal
English, there are a number of alternative ways to phrase the statement “if p, then q”, e.g.
e.g. “p implies q” or “q follows from p”.
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The easiest way to remember the right output values for this operation is to remember
that the value is false in exactly one case: when p is true and q is false.
Converse, Contrapositive, Biconditional
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The converse of p
q is q
p. The two statements are not equivalent
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The converse mostly occurs in two contexts. First, getting the direction of implication
backwards is a common bug in writing proofs. That is, using the converse rather than the
original statement. This is a bug because implications frequently hold in only one
direction.
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Second, the phrase “p implies q, and conversely” means that p and q are true under
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 Spring '08
 Erickson
 Computer Science

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