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Unformatted text preview: both open sets then X Y is an open set. f (0 + ) - f (0) f (0 + ) - f (0) lim 0- the derivative does not exist at x = 0.
0+ 0 + - 0 epsilon = lim = -1 0- lim 7. Let P and Q be atomic propositions. The propositional form [(P Q) P ] P is a tautology. Solution: The truth-table is: P T T F F Q [(P Q) P ] P T T F T T T F T Solution: True. If X is open, then for every x X, > 0 so that B (x) X. Similarly, if Y is open then for every x Y , > 0 so that B (x) Y . Therefore, take any x X Y . We need to show that > 0 so that B (x) X Y . But this is straightforward since x X Y implies x X or x Y . If the former is true, then B (x) X X Y . If the latter is true then B (x) Y X Y . Hence, X Y is open. Therefore, by definition, it is a tautology. 8. Let P , Q and R be atomic propositions. The propositional form (P Q) (R R) is equivalent to the propositional form [ R (P Q)] R Solution: The truth-tables are: P T...
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This document was uploaded on 03/20/2014 for the course ECON 2p30 at Brock University, Canada.
- Fall '12