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:

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

$$
LATEXBLOCK0
$$

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.

Answer:

• (A) Yes, it is valid.
• (B) No, it is not valid. (Correct answer)
• (C) It cannot be determined.