Question

1. translate the argument into symbolic form and use a truth table to determine whether the argument is valid or invalid.

if i exercise or go on a diet, then i lose weight.
i lost weight.

therefore, i exercised or went on a diet.

\

$$\begin{array}{|c|c|c|c|c|c|} \\hline p & q & r & & & \\\\ \\hline t & t & t & & & \\\\ \\hline t & t & f & & & \\\\ \\hline t & f & t & & & \\\\ \\hline t & f & f & & & \\\\ \\hline f & t & t & & & \\\\ \\hline f & t & f & & & \\\\ \\hline f & f & t & & & \\\\ \\hline f & f & f & & & \\\\ \\hline \\end{array}$$

a. yes, it is valid.
b. no, it is not valid.
c. it cannot be determined.

Explanation:

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<reasoning>

Define simple statements

Using the Mathematical Logic knowledge point
Let:

• \(p\): I exercise.
• \(q\): I go on a diet.
• \(r\): I lose weight.

Translate the argument

Using the Mathematical Logic knowledge point

• Premise 1: If I exercise or go on a diet, then I lose weight.

\[(p \lor q)
ightarrow r\]

• Premise 2: I lost weight.

\[r\]

• Conclusion: Therefore, I exercised or went on a diet.

\[p \lor q\]

Construct the truth table

Using the Truth Tables knowledge point
\[

$$\begin{array}{ccc|c|c|c} p & q & r & p \lor q & (p \lor q) ightarrow r & \text{Premises: } (p \lor q) ightarrow r \text{ and } r \\ \hline T & T & T & T & T & \text{Both True} \\ T & T & F & T & F & \\ T & F & T & T & T & \text{Both True} \\ T & F & F & T & F & \\ F & T & T & T & T & \text{Both True} \\ F & T & F & T & F & \\ F & F & T & F & T & \text{Both True} \\ F & F & F & F & T & \end{array}$$

\]

Evaluate argument validity

Using the Truth Tables knowledge point

• Critical rows where both premises \((p \lor q)

ightarrow r\) and \(r\) are True:

• Row 1: \(p=T, q=T, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 3: \(p=T, q=F, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 5: \(p=F, q=T, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 7: \(p=F, q=F, r=T\). Conclusion \(p \lor q = F\) (Invalid).
• Since the conclusion is False in Row 7 while all premises are True, the argument is invalid.

</reasoning>

<answer>
<mcq-option>(A) Yes, it is valid.</mcq-option>
<mcq-correct>(B) No, it is not valid.</mcq-correct>
<mcq-option>(C) It cannot be determined.</mcq-option>
</answer>

<post_analysis>
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</post_analysis>

Answer:

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<reasoning>

Define simple statements

Using the Mathematical Logic knowledge point
Let:

• \(p\): I exercise.
• \(q\): I go on a diet.
• \(r\): I lose weight.

Translate the argument

Using the Mathematical Logic knowledge point

• Premise 1: If I exercise or go on a diet, then I lose weight.

\[(p \lor q)
ightarrow r\]

• Premise 2: I lost weight.

\[r\]

• Conclusion: Therefore, I exercised or went on a diet.

\[p \lor q\]

Construct the truth table

Using the Truth Tables knowledge point
\[

$$\begin{array}{ccc|c|c|c} p & q & r & p \lor q & (p \lor q) ightarrow r & \text{Premises: } (p \lor q) ightarrow r \text{ and } r \\ \hline T & T & T & T & T & \text{Both True} \\ T & T & F & T & F & \\ T & F & T & T & T & \text{Both True} \\ T & F & F & T & F & \\ F & T & T & T & T & \text{Both True} \\ F & T & F & T & F & \\ F & F & T & F & T & \text{Both True} \\ F & F & F & F & T & \end{array}$$

\]

Evaluate argument validity

Using the Truth Tables knowledge point

• Critical rows where both premises \((p \lor q)

ightarrow r\) and \(r\) are True:

• Row 1: \(p=T, q=T, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 3: \(p=T, q=F, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 5: \(p=F, q=T, r=T\). Conclusion \(p \lor q = T\) (Valid).
• Row 7: \(p=F, q=F, r=T\). Conclusion \(p \lor q = F\) (Invalid).
• Since the conclusion is False in Row 7 while all premises are True, the argument is invalid.

</reasoning>

<answer>
<mcq-option>(A) Yes, it is valid.</mcq-option>
<mcq-correct>(B) No, it is not valid.</mcq-correct>
<mcq-option>(C) It cannot be determined.</mcq-option>
</answer>

<post_analysis>
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