By Smith D., Eggen M., Andre R.
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Extra resources for A transition to advanced mathematics
This method is valid because of the tautology [(P ∨ Q) ⇒ R] ⇐ ⇒ [(P ⇒ R) ∧ (Q ⇒ R)]. ” The two similar statement forms (P ⇒ Q) ⇒ R and P Q (Q Q R) have remarkably dissimilar direct proof outlines. For (P ⇒ Q) ⇒ R, we assume P ⇒ Q and deduce R. We cannot assume P; we must assume P ⇒ Q. On the other hand, in a direct proof of P ⇒ (Q ⇒ R), we do assume P and show Q ⇒ R. Furthermore, after the assumption of P, a direct proof of Q ⇒ R begins by assuming Q is true as well. This is not surprising since P ⇒ (Q ⇒ R) is equivalent to (P ∧ Q) ⇒ R.
Show that ( Ex)(x 7 − 12x 3 + 16x − 3 = 0) is true in the universe of real numbers. For the polynomial f (x) = x 7 − 12x 3 + 16x − 3, f (0) = −3 and f (1) = 2. From calculus, we know that f is continuous on [0, 1]. The Intermediate Value Theorem tells us there is a zero for f between 0 and 1. Even if we don’t know the exact value of the zero, we know it exists. Therefore, the truth set of x 7 − 12x 3 + 16x − 3 = 0 is nonempty. Hence (Ex)(x 7 − 12x 3 + 16x − 3 = 0) is true. ” To decide the truth value of the given sentence in the universe ގit is not enough to observe that 32 > 3, 42 > 3, and so on.
The converse of P ⇒ Q is Q ⇒ P. The contrapositive of P ⇒ Q is (∼Q ) ⇒ (∼P). ” The converse is false, but the sentence and its contrapositive are true. ” In this example, all three sentences are true. The previous two examples suggest that the truth values of a conditional sentence and its contrapositive are related, but there seems to be little connection between the truth values of P ⇒ Q and its converse. We describe the relationships in the following theorem. 1 For propositions P and Q, (a) (b) P ⇒ Q is equivalent to its contrapositive (∼Q) ⇒ (∼P).
A transition to advanced mathematics by Smith D., Eggen M., Andre R.