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Unformatted text preview: Math 6a - Solution Set 3 Problem # 1 Story problem 1 My question for you, dear Carib, is this: Can you find another truth-functional connective that is associative? That is, is there some well defined way of taking two truth values, p and q, and producing a third-call it p q-so that ( p q ) r = p ( q r ) for any choice of truth values (true or false) p,q, and r ? Solution. We can simply define p q = T for all possible truth values of p and q. Then, trivially, both ( p q ) r and p ( q r ) both are always equal to T no matter what value p, q, and r have. Problem # 2 Story problem 2 Heres another question for you: Which of these connectives have inverses ? (Every real number p other than 0 has a multiplicative inverse, that is, a number p- 1 such that p p- 1 = 1 . ) In other words, if p is an arbitrary truth-value (i.e., true or false), can we always find a truth- value-call it p- 1-such that p p- 1 = F Or p & p- 1 = T Or p p- 1 = T ? Solution. In the first case, we cannot find an inverse in general, since if p = T there is no truth value q such that p q = T q = F. Similarly, in the second case, we again cannot find an inverse in general, since if p = F there is no truth value q such that p & q = F & q = T. In the third case, we can find an inverse in general. Simply let p- 1 = p. Then, if p = T we have p p- 1 = p p = T T = T and similarly, if p = F, we have p p- 1 = p p = F F = T. Problem # 3 Story problem 3 Prove that distributes over : i.e., that p ( q r ) = ( p q ) ( p r ) 1 2 for all truth-values p,q, and r. Solution. First, note that if p = T, the p ( q r ) = T no matter what q and r are, and similarly, p q and p r are both T and thus ( p q ) ( p r ) = T T = T and thus the left side equals the right side. So consider the case when p = F. Then, p x is true if x is true, and false if x is false, and thus, in either case p x = x. Thus, we can simplify the right hand side p ( q r ) = q r and the left hand side ( p q ) ( p r ) = q r and thus, again, both sides are equal, no matter what q and r are. This covers all cases. Problem # 4 Story problem 4 Well, his majesty said at last, tell me this: for what truth-values x is ( x x ) = F true? Solution. There are only two possible truth values, T or F. When x = T, x x = T T = T so this is not a solution to the above equation. When x = F, x x = F F = F so x = F is a solution to the above equation. Thus, the only solution is x = F. Problem # 5 Story problem 5 Thats an interesting question, Abubakari said (aloud), once hed taken it all in. He was quite enjoying this. Can we find a polynomial f ( x ) with coefficients in L-that is, with coefficients that are 0 or 1-which has no root? That is, such that f ( x ) is never zero, no matter what x is? Solution. Consider the polynomial...
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- Fall '07