2. Constructing a Truth Table The number of lines in a truth table for a compoun

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2. Constructing a Truth Table The number of lines in a truth table for a compound proposition is determined by the number of different simple statement components contained within that compound proposition. The number of required rows is determined by the exponential formula L 2, where L is the number of lines (rows) required and n is the number of different simple statements. A statement letter that occurs more than once represents a single unique statement and should be counted only once when you are determining the number of different simple statements. For example, the proposition z·(2 v F) contains only two different statement letters (Z and F), although statement Z occurs twice. Since there are only two different simple statements, the truth table for this proposition requires 2 4 lines. Once you know the number of required lines, go to the column beneath the first letter and write a T (for "true") in the first half of the rows and an F (for "false") in the second half of the rows. For example, if the truth table requires 8 total rows, go to the column beneath the first letter and write T in rows 1 through 4 and write F in rows 5 through 8. Then move over to the next unique letter to the right (skip any occurrences of the same letter), and cut the number of Ts and Fs in half. For example, if the column beneath the first letter has four Ts and four Fs, alternate between two Ts and two Fs in the column beneath the second unique letter until you get to the bottom of the column. Repeat this process until you have exhausted all the unique letters, reducing the number of alternating Ts and Fs by half every time you move to the column beneath a new unique letter. If you follow this method, you will ensure that every possible combination of truth values appears on the truth table. For each given compound proposition, indicate the number of unique statement letters the proposition has and how many truth table lines are required for that proposition. Begin the truth table for each compound proposition by following the method outlined above to assign the initial truth values beneath the first instance of each simple statement letter. Type a T or an F into the spaces provided. 0 The statement R E (RA R) has unique statement letter(s). Therefore, its truth table must have row(s).

Explanation / Answer

In the first truth table, the value of the 1st row is T and the value of the 2nd row is F.
The statement R.. has 1 unique letter.
Therefore, it's truth table must have 2 rows.
In the next table, the values in the first column (J) should be T, T, F, F. The values under the second column (S) should be T, F, T, F.

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