In words, the statement says that every prime is odd, and $2$ is clearly a counterexample to that statement. The statement $$\forall x\big(P(x)\to Q(x)\big)$$ is false, because $2$ is prime (i.e., $P(2)$ is true), but $2$ is not odd (i.e., $Q(2)$ is false). Suppose that the domain of discourse is the set of positive integers, $P(x)$ is the statement that $x$ is prime, and $Q(x)$ is the statement that $x$ is odd. Authentication Codes and Combinatorial Designs (Discrete Mathematics and Its Applications) Dingyi Pei BookSee - Download books for free.
$\forall xP(x)\to\forall xQ(x)$, on the other hand, says that if every $x$ has property $P$, then every $x$ also has property $Q$.
Then $\forall x\big(P(x)\to Q(x)\big)$ is true: if some integer $x$ is a multiple of $4$, then $x$ is certainly even.
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Suppose that we’re talking strictly about integers, $P(x)$ means that $x$ is a multiple of $4$, and $Q(x)$ means that $x$ is even. Sell Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games - ISBN 9780471476023 - Ship for free - Bookbyte This website uses cookies to ensure you get the best experience on our website. Now, with Douglas Ensley and Winston Crawleys Introduction to Discrete Mathematics, you can explore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems. In words, $\forall x\big(P(x)\to Q(x)\big)$ says that no matter what $x$ you take, if it has property $P$, then it also has property $Q$. File Type PDF Discrete Mathematics Solutions.
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