Use truth tables to decide the equivalence of each of the following pairs of expressions.  Remember that with three variables your truth tables will have to have eight lines.

1.  P -> (Q v R)  and  (P v Q) -> R
P
 Q
 R
 P -> (Q v R)
 (P v Q) -> R
1
 1
 1
 1
 1
1
 1
 0
 1
 0
1
 0
 1
 1
 1
1
 0
 0
 0
 0
0
 1
 1
 1
 1
0
 1
 0
 1
 0
0
 0
 1
 1
 1
0
 0
 0
 1
 1

no, not equivalent

2.  P -> (Q v R)  and  (P & Q) -> R
P
 Q
 R
 P -> (Q v R)
 (P & Q) -> R
1
 1
 1
 1
 1
1
 1
 0
 1
 0
1
 0
 1
 1
 1
1
 0
 0
 0
 1
0
 1
 1
 1
 1
0
 1
 0
 1
 1
0
 0
 1
 1
 1
0
 0
 0
 1
 1

no, not equivalent

3.  P -> (Q & R)  and  (P & Q) -> R
P
 Q
 R
 P -> (Q & R)
 (P & Q) -> R
1
 1
 1
 1
 1
1
 1
 0
 0
 0
1
 0
 1
 0
 1
1
 0
 0
 0
 1
0
 1
 1
 1
 1
0
 1
 0
 1
 1
0
 0
 1
 1
 1
0
 0
 0
 1
 1

no, not equivalent

4.  P -> (Q & R)  and  (P v Q) -> R
P
 Q
 R
 P -> (Q & R)
 (P v Q) -> R
1
 1
 1
 1
 1
1
 1
 0
 0
 0
1
 0
 1
 0
 1
1
 0
 0
 0
 0
0
 1
 1
 1
 1
0
 1
 0
 1
 0
0
 0
 1
 1
 1
0
 0
 0
 1
 1

no, not equivalent

5.  P -> (Q v R)  and  (~P v Q) v R
P
 Q
 R
 P -> (Q v R)
 (~P v Q) v R
1
 1
 1
 1
 1
1
 1
 0
 1
 1
1
 0
 1
 1
 1
1
 0
 0
 0
 0
0
 1
 1
 1
 1
0
 1
 0
 1
 1
0
 0
 1
 1
 1
0
 0
 0
 1
 1

yes, equivalent!  This involves an example of what we will soon meet as the material equivalence rule.

6.  P -> (Q & R)  and  (~P v Q) & (~P v R)
P
 Q
 R
 P -> (Q & R)
 (~P v Q) & (~P v R)
1
 1
 1
 1
 1
1
 1
 0
 0
 0
1
 0
 1
 0
 0
1
 0
 0
 0
 0
0
 1
 1
 1
 1
0
 1
 0
 1
 1
0
 0
 1
 1
 1
0
 0
 0
 1
 1

yes, equivalent!  This is an example of what we will call the distribution rule.

7.  P -> (Q -> R)  and  (P & Q) -> R
P
 Q
 R
 P -> (Q -> R)
(P & Q) -> R
1
 1
 1
 1
 1
1
 1
 0
 0
 0
1
 0
 1
 1
 1
1
 0
 0
 1
 1
0
 1
 1
 1
 1
0
 1
 0
 1
 1
0
 0
 1
 1
 1
0
 0
 0
 1
 1

yes, equivalent!  This is an example of what we will call the exportation rule.