Question

Let $S=\{1,2,3\}$. Consider the following open sentences over the domain $S$ : $P(n): \frac{(n+4)(n+5)}{2}$ is odd. $$ Q(n): 2^{n-2}+3^{n-2}+6^{n-2}>(2.5)^{n-1} . $$ Determine three distinct elements $a, b, c$ in $S$ such that $P(a) \Rightarrow Q(a)$ is false, $Q(b) \Rightarrow P(b)$ is false and $P(c) \Leftrightarrow Q(c)$ is true.

   Let $S=\{1,2,3\}$. Consider the following open sentences over the domain $S$ :
$P(n): \frac{(n+4)(n+5)}{2}$ is odd.
$$
Q(n): 2^{n-2}+3^{n-2}+6^{n-2}>(2.5)^{n-1} .
$$

Determine three distinct elements $a, b, c$ in $S$ such that $P(a) \Rightarrow Q(a)$ is false, $Q(b) \Rightarrow P(b)$ is false and $P(c) \Leftrightarrow Q(c)$ is true.

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Mathematical Proofs: A Transition to Advanced Mathematics

Mathematical Proofs: A Transition to Advanced Mathematics

Gary Chartrand,… 4th Edition

Chapter 2, Problem 43

Instant Answer

Step 1

- For \( n = 1 \): \[ P(1): \frac{(1+4)(1+5)}{2} = \frac{5 \times 6}{2} = 15 \quad \text{(odd)} \] So, \( P(1) \) is true. - For \( n = 2 \): \[ P(2): \frac{(2+4)(2+5)}{2} = \frac{6 \times 7}{2} = 21 \quad \text{(odd)} \] So, \( P(2) \) is Show more…

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Let $S=\{1,2,3\}$. Consider the following open sentences over the domain $S$ : $P(n): \frac{(n+4)(n+5)}{2}$ is odd. $$ Q(n): 2^{n-2}+3^{n-2}+6^{n-2}>(2.5)^{n-1} . $$ Determine three distinct elements $a, b, c$ in $S$ such that $P(a) \Rightarrow Q(a)$ is false, $Q(b) \Rightarrow P(b)$ is false and $P(c) \Leftrightarrow Q(c)$ is true.

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