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# Write the given argument in words and determine whether each argument is valid. Let p: 4 gigabytes is better than no memory at all, q: We will buy more memory, r: We will buy a new computer.$\begin{array}{l} p\rightarrow r \\ r\rightarrow q \\ p \\ \hline \therefore q \end{array}$

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Given:

$p$: 4 gigabytes is better than no memory at all

$q$: We will buy more memory.

$r$: We will buy a new computer.

$\begin{matrix} \\ \\ \therefore\end{matrix} \: \: \: \: \: \begin{matrix} p\rightarrow r \\ r\rightarrow q \\ p \\ \hline q\end{matrix}$

$\textbf{Conditional statement}$ $p\rightarrow q$: if $p$, then $q$

$\begin{matrix} \\ \\ \therefore\end{matrix} \: \: \: \: \: \begin{matrix} \text{If p, then r} \\ \text{If r, then q} \\ p \\ \hline q \end{matrix}$

Replace "4 gigabytes is better than no memory at all" by $p$, "We will buy more memory." by $q$, and "We will buy a new computer." by $r$.

$\begin{matrix} \\ \\ \therefore\end{matrix} \: \: \: \: \: \begin{matrix} \text{If 4 gigabytes is better than no memory at all, then we buy a new computer.} \\ \text{If we buy a new computer, then we buy more memory.} \\ \text{4 gigabytes is better than no memory at all} \\ \hline \text{We buy more memory.} \end{matrix}$

Let us assume that $p\rightarrow r$, $r\rightarrow q$ and $p$ are true.

Since $p$ and $p\rightarrow r$ are both true, we require that $r$ is true as well.

Since $r$ and $r\rightarrow q$ are both true, we require that $q$ is true as well.

When $p\rightarrow r$, $r\rightarrow q$ and $p$ are true, $q$ is true as well and thus the argument is valid.

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