SOLUTION: Determine which of the following three statements are equivalent. (a) not r or ( not q and not r) (b) not r or (not p and not q) (c) not[ r or ( p and q) ]

Notation ~ means "not", \/ means "or", /\ means "and"

Rules:  ~ means to change whatever follows it to F if it is T
          and to T if it is F

       /\ is F unless it has T on both sides of it; then it's T

       \/ is T unless it has F on both sides of it; then it's F 
 
Make truth tables for all three following the above rules:

Truth table for the first one: 

(a) not r or ( not q and not r)

Note: p does not appear in this one and hence a
4-line truth table would do, but we will put in
a column for p anyway so we can easily compare the final
result with the results of the other two.

p|q|r|~q|~r|~q/\~r|~r\/(~q/\~r)
T|T|T| F| F|   F  |   F  
T|T|F| F| T|   F  |   T
T|F|T| T| F|   F  |   F
T|F|F| T| T|   T  |   T
F|T|T| F| F|   F  |   F
F|T|F| F| T|   F  |   T
F|F|T| T| F|   F  |   F
F|F|F| T| T|   T  |   T

(b) not r or (not p and not q)

p|q|r|~p|~q|~r|~p/\~q|~r\/(~p/\~q)
T|T|T| F| F| F|   F  |   F
T|T|F| F| F| T|   F  |   T
T|F|T| F| T| F|   F  |   F
T|F|F| F| T| T|   F  |   T
F|T|T| T| F| F|   F  |   F
F|T|F| T| F| T|   F  |   T
F|F|T| T| T| F|   T  |   T
F|F|F| T| T| T|   T  |   T

(c) not[ r or ( p and q) ]

p|q|r|p/\q|r\/(p/\q)|~[r\/(p/\q)]
T|T|T|  T |  T      |    F
T|T|F|  T |  T      |    F
T|F|T|  F |  T      |    F
T|F|F|  F |  F      |    T
F|T|T|  F |  T      |    F
F|T|F|  F |  F      |    T
F|F|T|  F |  T      |    F
F|F|F|  F |  F      |    T

None of the three have the exact same truth table, so 
none of them are equivalent.

Edwin