If P then Q

Date: 08/29/97 at 03:40:07
From: Harout Jarchafjian
Subject: If p then q

Our math book states that the implication of if p then q, the truth 
table, is

  p = true,  q = true,  statement = true
  p = true,  q = false, statement = false
  p = false, q = doesn't matter, statement = true 

I don't understand how if p is false then regardless of q, the 
statement is true. My math book gives no examples of this. Can you 
please give an example?

Thank you,
Harout Jarchafian

Date: 08/29/97 at 19:01:09
From: Doctor Anthony
Subject: Re: If p then q

p implies q can be written    p -> q

This means that p is a subset of q.

Draw a venn diagram with the set p completely enclosed within set q.  
Draw a box enclosing set q to represent the universal set.

Then the area inside both p and q is  pq

  area outside p but still inside q is  p'q

  area outside q but inside the box is  p'q'

Then 1 represents the universal set and adding the three areas gives 
the universal set, so

       pq + p'q + p'q' = 1

Now draw up the truth table for  p -> q

       p    q      p -> q
      ---------------------
       0    0        1         This line is p'q'
       0    1        1         This line is p'q 
       1    0        0         This line is pq'  (does not contribute)
       1    1        1         This line is pq
      ----------------------

From this table you can see that p=0 (i.e. p'=1) always gives p->q = 1

This is equivalent to saying p = false means p->q is always true.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/