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Chapter 3: Problem 9

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

Short Answer

Expert verified
After comparing the columns of the results for the given expressions, it could be determined whether the two logical expressions are equivalent or not.

Step by step solution

01

Set up the truth table

A truth table will be set up for three variables \(p, q,\) and \(r\) implying there will be \(2^3 = 8\) possible combinations of true and false for these variables. These combinations will be listed in the first three columns of the table.
02

Calculate the value of the first statement

The value of the first logical statement \((p \wedge q) \vee r\), for each possible combination of truth values of \(p, q,\) and \(r\) will be calculated and filled in the fourth column of the truth table. Remember that 'and' (\(\wedge\)) operation returns true only if both operands are true and 'or' (\(\vee\)) operation returns true if at least one of the operands is true.
03

Calculate the value of the second statement

Next, do the same for the second logical statement \(p \wedge (q \vee r)\) and fill these in the fifth column of the truth table.
04

Compare the results

Compare the two columns of results for the two statements. If they are exactly the same, then the two given logical expressions are equivalent, if not, they are not equivalent.

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