Constructing Truth Tables

Date: 02/17/2006 at 22:48:45
From: Yvonne
Subject: truth tables

Construct a truth table for the statement ~q v (p^r).

I understand how to put it in symbolic form but not how to construct 
the truth table for it.  I know what the symbols mean but am confused
on why one is true and one is false.

Date: 02/18/2006 at 13:48:13
From: Doctor Achilles
Subject: Re: truth tables

Hi Yvonne,

Thanks for writing to Dr. Math.

First, let's just make sure that we are both using the same symbols.

  "~" is a negation.  ~X is true when X is false, and false when X is
true.

  "v" means "or".  (X v Y) is true when X is true (no matter what Y
is).  It is also true when Y is true (no matter what X is).  The only
way it is false is if *both* X *and* Y are false.

  "^" means "and".  (X ^ Y) is false when X is false (no matter what Y
is).  It is also false when Y is false (no matter what X is).  The
only way it is true is if *both* X *and* Y are true.


Here's how I make a truth table.  I make 1 column for each variable. 
In your case, the two variables are p and q.  Then I make another
column for the whole sentence.  So the columns for your first truth
table are:

  p    q    r     (~q v (p ^ r))

Then, I list all the possible combinations of True and False for each
variable.  For three variables, the possibilities are:

  True, True, True
  True, True, False
  True, False, True
  True, False, False
  False, True, True
  False, True, False
  False, False, True
  False, False, False

So our truth table needs to have 8 lines.

So now the table looks like this:

  p    q    r     (~q v (p ^ r))
 ---  ---  ---    --------------
  t    t    t         ?
  t    t    f         ?
  t    f    t         ?
  t    f    f         ?
  f    t    t         ?
  f    t    f         ?
  f    f    t         ?
  f    f    f         ?

Next, I go row by row and fill in the table.

If p is true and q is true and r is true, then you have:

  (~true v (true ^ true))

The first thing we do is work out the (true ^ true) in the
parentheses, this leaves us with:

  (~true v true)

Next, we work out the "~true":

  (false v true)

Finally, we work out the "v".  (false v true) is true, so we have:

  true

So our truth table is:

  p    q    r     (~q v (p ^ r))
 ---  ---  ---    --------------
  t    t    t         t
  t    t    f         ?
  t    f    t         ?
  t    f    f         ?
  f    t    t         ?
  f    t    f         ?
  f    f    t         ?
  f    f    f         ?

Next we have:

  (~true v (true ^ false))

Using the same steps, we get:

  (~true v false)

  (false v false)

  false

So now we have:

  p    q    r     (~q v (p ^ r))
 ---  ---  ---    --------------
  t    t    t         t
  t    t    f         f
  t    f    t         ?
  t    f    f         ?
  f    t    t         ?
  f    t    f         ?
  f    f    t         ?
  f    f    f         ?

Try to fill in the rest of the table yourself.  If you get stuck, just
write back and let me know where you ran into trouble.

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/