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/