PDNF and PCNF in Discrete Mathematics

In Discrete Mathematics, PDNF and PCNF are standard ways of expressing a Boolean function. In these forms, each term contains all variables exactly once. PDNF represents the function using minterms corresponding to output 1, whereas PCNF uses maxterms corresponding to output 0.

PDNF (Principal Disjunctive Normal Form)

PDNF is a logical expression written in the Sum of Products (SOP) form where every product term contains all variables exactly once, either in normal form or complemented form. The terms are joined using the OR (+) operator.

Example: (P . Q' . R) + (P' . Q . R) + (P . Q . R')

Here, the ‘+’ sign represents the OR operation, and each term contains all variables P, Q, and R.

PCNF (Principal Conjunctive Normal Form)

PCNF is a logical expression written in the Product of Sums (POS) form where every sum term contains all variables exactly once, either in normal form or complemented form. The terms are joined using the AND (.) operator.

Example: (P + Q' + R) . (P' + Q + R) . (P + Q + R')

Here, the ‘.’ sign represents the AND operation, and each term contains all variables P, Q, and R.

Properties

1. Every PDNF or PCNF corresponds to a unique Boolean Expression and vice versa.
2. If X and Y are two Boolean expressions, then X is equivalent to Y if and only if PDNF(X) = PDNF(Y) or PCNF(X) = PCNF(Y).
3. For a Boolean Expression, if PCNF has m terms and PDNF has n terms, then the number of variables in such a Boolean expression = .

Solved Examples

Example 1: Convert the following Boolean expression to PDNF: A . (B + C')

Solution:

1. Distribute A

A . B + A . C'

2.Ensure each product term contains all variables

(A . B . C') + (A . B . C) + (A . B' . C')

Example 2: Convert the following Boolean expression to PCNF: (A + B') . (B + C)

Solution:

1.Ensure each sum term contains all variables using Boolean identity

(A+B′) = (A+B′+C) . (A+B′+C′)

(B+C) = (A+B+C) . (A′+B+C)

2.Write the expression in PCNF

(A+B′+C) . (A+B′+C′) . (A+B+C) . (A′+B+C)

Practice Problems

1. Convert (A + B) . (B' + C) to PCNF.
2. Convert A . (B + C) + A' . B . C' to PDNF.
3. Simplify (P . Q' . R) + (P' . Q) to PDNF.
4. Express (X + Y') . (Y + Z') . (X + Z) in PCNF.
5. Convert (M . N + M' . N') . (P + Q') to PDNF.
6. Simplify (P + Q) . (Q' + R) . (R + P') to PCNF.
7. Express A . (B + C) + A' . (B . C') in PDNF.
8. Convert (A + B) . (B + C) . (C + A') to PCNF.
9. Simplify P . (Q + R') + Q' . R to PDNF.
10. Convert (X + Y) . (Y + Z) . (Z + X') to PCNF.