This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 truthfunctional connective that is associative? That is, is there some well defined way of taking two truth values, p and q, and producing a thirdcall it p qso 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 truthvalue (i.e., true or false), can we always find a truth valuecall it p 1such 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 truthvalues 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 truthvalues 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 Lthat is, with coefficients that are 0 or 1which has no root? That is, such that f ( x ) is never zero, no matter what x is? Solution. Consider the polynomial...
View Full
Document
 Fall '07
 Wilson
 Math

Click to edit the document details