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Unformatted text preview: scope
In order to parse a logical expression we need to know which subexpressions to parse rst. Although there's often conventions (just as in your favourite programming language), such as evaluating before A, when in doubt you should use parentheses:
P x ( ) Q(x) A R(x) : (P (x) Q(x)) A R(x) versus P x ( ) (Q(x) A R(x)) This becomes particularly important when you want to be explicit about the scope of universal quantiers: (Vx P R; Wy P R; x < y ) A (Vx P R; Wy P R; x2 < y ) Notice that the scope of the quantication is inside the relevant parentheses. There's no reason that the y in the antecedent would be the same as the y in the consequent. It could be rewritten: (Vx P R; Wy P R; x < y ) A (Vz P R; Ww P R; z 2 < w) slide 13 drawing truth As conjunctions, disjunctions, negations, and other combinations of predicates become more ornate, we need help to interpret them. To think about the truth value of up to three predicates, you can probably draw a venn diagramm. For example, draw the venn diagram showing which region(s) could not have any elements, and still remain consistent with P (x) A (Q(x) A R(x)). How would you draw the analogous diagram for predicates P; Q; R, and S ? Perhaps if your 3D rendering skills were pretty good you'd manage. However, to combine more predicates, you need a new tool. slide 14 tabulating truth
The standard venn diagram for 3 sets has 23 regions  one region for each possible combination of truth values for its component sets. We can get the same eect with a rectangular diagram, or table: P T T T T F F F F Q T T F F T T F F R T F T F T F T F Q A
T F T T T F T T R P A( A
Q T F T T T T T T R ) As an exercise, compare this to the table for (P Q ) A R. What do you conclude? slide 15 ...
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This note was uploaded on 10/13/2011 for the course COMPUTER S CSC 165 taught by Professor Dannyheap during the Fall '10 term at University of Toronto Toronto.
 Fall '10
 DannyHeap
 Computer Science

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