Constructing Truth Tables
Date: 11/03/2000 at 06:11:03 From: Amber Zertuche Subject: Mathematical truth tables Dr. Math, I am doing a lesson on truth tables and propositional logic. When there is a question with multiple possible answers, how do you write it in the table? For example, for pv(p^-q), how would I state this as true or false? Are there several ways it can be true or false?
Date: 11/03/2000 at 11:29:15 From: Doctor TWE Subject: Re: Mathematical truth tables Hi Amber - thanks for writing to Dr. Math. I teach truth tables too (more often as they relate to digital electronic circuits, but the principles are the same), and I have the students construct their truth tables in three parts: the inputs, the "partial outputs," and the final output(s). For your example (I'm reading it as "p or the quantity p and not q"), the inputs would be p and q. With two inputs, there are 2^2 = 4 combinations (for n inputs, there are always 2^n combinations), and they're traditionally listed like this: p q | -----+- t t | t f | f t | f f | I have the students put a thick vertical line between the inputs and the partial outputs. Next, I have them add a column for each partial output. Partial outputs are all the intermediate results needed to determine the final output. In your example, we'll need a partial output column for ~q (I prefer the ~ for logical negation) and another for p^~q. Adding these, we have: p q | ~q p^~q | -----+----------+ t t | | t f | | f t | | f f | | Again, I have the students separate the partial outputs and the final output with a thick vertical line. Then, of course, we'll add a column for the final output: p q | ~q p^~q | pv(p^~q) -----+----------+---------- t t | | t f | | f t | | f f | | Now the students need to fill in the table. I recommend that they proceed down the columns instead of going across the rows. That way they only have to deal with one logic operation at a time, and they're always using the same input columns. In the example, we'd first fill in the ~q column by looking at the each value in the q input column and entering its complement: p q | ~q p^~q | pv(p^~q) -----+----------+---------- t t | f | t f | t | f t | f | f f | t | Next, we'll fill in the p^~q column. In each row, we'll look at the values in the p column and the ~q column (which we just filled in), and AND them together. So for the first row, p is "t" and ~q is "f" so p^~q is "t" AND "f", which is "f". We enter that: p q | ~q p^~q | pv(p^~q) -----+----------+---------- t t | f f | t f | t | f t | f | f f | t | In the second row, p is "t" and ~q is "t" so p^~q is "t" AND "t", which is "t". We continue down the column and we have: p q | ~q p^~q | pv(p^~q) -----+----------+---------- t t | f f | t f | t t | f t | f f | f f | t f | Finally, we need to figure out the final output column. The "inputs" for the last operation (the OR) are p and p^~q, so we'll look at the values in these two columns (row by row) as we did in the previous step, and get our final output column values. We then have our finished truth table: p q | ~q p^~q | pv(p^~q) -----+----------+---------- t t | f f | t t f | t t | t f t | f f | f f f | t f | f I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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