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Unformatted text preview: Homework 1 Solutions 1. Construct a truth table for the statement: [ p ( q q )] p. Solution: First note that q q is a contradiction, so replace it with the variable c which only takes on the truth value F. Then we get the truth table: p c p c p [ p c ] p [ p c ] p [ p c ] p T F F F T T T F F T T T T T Because the truth values of the statement are always T, this says the statement is a tautology. In other words, whenever you want to prove that p c where c is always F (i.e. c is a contradiction), you can instead prove p , and vice-versa. 2. Consider the following statement: A function f is uniformly continuous on a set S if and only if for every > there is a > such that | f ( x )- f ( y ) | < whenever x and y are in S and | x- y | < . (a) Rewrite the statement using the logical symbols , , , , etc. Solution: [ f is uniformly continuous on a set S ] [ > 0, > 0 so that x,y S and | x- y | < | f ( x )- f ( y ) | .] (b) Write the negation of the statement in words and in logical symbols....
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