Question 162438This question is from textbook
: problem 1f:
-(PvQ) -||- -P&-Q
This question is from textbook
Answer by vleith(2983)   (Show Source):
You can put this solution on YOUR website! One way to do all these logic type problems is to make a table. Make two rows for each variable (in this case 4 rows, one for P and one for Q). Then make a column for each logical step you take. The spacing will be messed up, but you can line them up by taking out all the fillers (+).
Start by adding both true (1) and false (0) for each variable. Do this until you have each combination for all variables. Since we have two variables, that makes 4 total combinations (2 * 2). See the table below
+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)
+0++0
+1++0
+0++1
+1++1
Now, using the info about P and Q, you can easily find (-P) and (-Q). Fill them in as follows.
+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)
+0++0+++++1++++++1+
+1++0+++++0++++++1+
+0++1+++++1++++++0+
+1++1+++++0++++++0+
Next, using logical properties for AND, OR, etc, continue completing the table Fill them in as follows.
+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)
+0++0+++++1++++++1+++++0++++++++1
+1++0+++++0++++++1+++++1++++++++0
+0++1+++++1++++++0+++++1++++++++0
+1++1+++++0++++++0+++++1++++++++0
Finally, using the logical negative, complete the table. Then note that the columns for the two entities in the original problem are identical
+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)
+0++0+++++1++++++1+++++0++++++++1+++++++++1
+1++0+++++0++++++1+++++1++++++++0+++++++++0
+0++1+++++1++++++0+++++1++++++++0+++++++++0
+1++1+++++0++++++0+++++1++++++++0+++++++++0
So what you need to know are:
The rules for ANA, OR, negation, implication, etc. Your book will have all that stuff. Then you need to be methodical about starting with a truth table and gradually working your way through each logical step. Once you get two columns that have the same entries, you have proven those columns are logical equivalents.
|
|
|
