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Chapter 3: Problem 10

Use a truth table to determine whether the two statements are equivalent. \((p \vee q) \wedge r, p \vee(q \wedge r)\)

Short Answer

Expert verified
The two logical statements are not equivalent. One can observe this from the truth table that displays at least one combination for which they evaluate to different truth values.

Step by step solution

01

Set up the Truth Table

To complete the truth table, we must consider all possible combinations of truth values for \(p\), \(q\), and \(r\). This gives us a total of \(2^3 = 8\) rows in our truth table, each corresponding to a unique combination of truth values for \(p\), \(q\), and \(r\).
02

Calculate the Truth Values for the two statements

We must calculate the truth values for each of the two statements \((p \vee q) \wedge r\) and \(p \vee(q \wedge r)\) for each combination of truth values for \(p\), \(q\), and \(r\). For \((p \vee q) \wedge r\), first compute \(p \vee q\), then 'AND' it with \(r\). For \(p \vee(q \wedge r)\), first compute \(q \wedge r\), then 'OR' it with \(p\).
03

Determine if Statements are Equivalent

In order for the two statements to be equivalent, they must have the same truth values for all combinations of truth values for \(p\), \(q\), and \(r\). This means that the columns for \((p \vee q) \wedge r\) and \(p \vee(q \wedge r)\) in the truth table must be equal. If there is at least one combination for which they are not equal, the two statements are not equivalent.
04

Conclusion

Finally, we conclude whether or not the two given logical statements are equivalent based on the truth table. If we found any discrepancies in the results from Step 3, we state that the two statements are not equivalent. If no discrepancies were found, we conclude that the statements are indeed equivalent.

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