truth table for the compound proposition

Question

¬p ∨ (r → ¬q)

 

Answer 1

Hi, Tevell!

 

Before I show you the truth table for the statement that you gave above, let me make sure that you understand all the parts of the statement.  If you have a statement p, the statement ~p is read as "not p" and understood to be the opposite of what p says.  The symbol 'v' represents an OR statement, which is only false is both parts are false.  In all other cases, the statement is true.  The symbol '→' represents an IF...THEN statement, which is only false when the first statement is true and the second is false.  In all other cases, the statement is true.  The compound proposition that you mentioned above is unique in that is has three individual statements (p, q, and r), so keep in mind that you need to represent all possible T/F combinations of p, q, and r in the first three columns (see below).

 

So, let's take a look at the truth table for the statement ~p v (r→~q):

p    q    r    ~p    ~q    r→~q    ~p v (r→~q) 

T     T   T     F     F        F             F

T     T   F     F     F        T             T

T     F   F     F     T        T             T

F     T   T     T     F        F             T

F     T   F     T     F        T             T

F     F   F     T     T        T             T

 

So the statement is true in all cases, except when p, q, and r are all true.  I hope this helps!

 

Patty