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Unformatted text preview: are both false. By inspection of the truth-table, this seems less likely to be possible since there is only one combination of truth values of p and q which makes p = q false. We now prove that it is, in fact, impossible to Fnd such propositions. Proof. By contradiction, suppose that there were propositions p and q such that p = q is false and q = p is also false. Since p = q is false, it must be the case that p is true and q is false (by the third line of the truth-table). Similarly, since q = p is false, q is true and p is false. Thus, p must be both true and false, which is impossible. We arrived at contradiction when we assumed that there were such propositions so we conclude that there are no propositions where the implication and its converse are both false....
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- Spring '06