The programs used earlier
to create truth tables for PLN expressions depended on the idea that every
WFF could be expressed with a unique mathematical equivalent in a modular
2 form. For instance, if P is either 1 or 0, then the value of PN
(~P) could be determined through the formula P + 1 (mod 2). The expression
PQA (P & Q) would represent the multiplication of the values for P
and Q.
This is the full list of
equivalents.
PN
(~P) = P + 1
PQA (P & Q) = PQ
PQM = P + Q
PQB (P <-> Q) = P + Q + 1
PQO (P v Q) = PQ + P + Q
PQC (P -> Q) = PQ + P + 1
QPC (Q -> P) = PQ + Q + 1
Note that this differs from Boolean algebra in that logical addition (as in P + Q) represents mutual exclusion.
Some of the mathematical
relationships worth noting are these:
P
+ P = 0
PP = P
For example, the WFF (P
& ~Q) v Q can be seen in algebraic notation as (P(Q
+ 1))Q + P(Q + 1) + Q, which reduces to
(PQ
+ P)Q + (PQ + P) + Q, then to PQ + PQ + PQ
+ P + Q, and finally to PQ + P + Q.
1. (P & Q) -> P
2. (P -> Q) v Q
3. ~(P <-> Q) &
P
4. P -> (Q & P)
5. ~(P & Q) &
Q
Computer
science majors using PC's may want to look at a simplified version of the
program used to create truth tables. Download and run the DOS
program http://www.internetlogic.org/drills/pln2.exe
then
look at the original code (in BASIC)
by downloading
http://www.internetlogic.org/drills/pln2.txt
One more thing to note
about any WFF, when expressed in algebraic notation, is that it will take
the form of aPQ+bP+cQ+d, and the letters a
through d express the parameters 1 or 0.
For this reason any WFF could also be written with just these numbers.
For instance, since (P & Q) -> Q could be rewritten as PQQ+PQ+1 and
then as just 1, the values of the parameters would be
a = 0
b = 0
c = 0
d = 1
and the series 0001 would
establish this WFF as a tautology.
Decide what would be the string corresponding to the following series.
1. 1101
2. 1011
3. 0010
4. 0100
5. 1000