Product of sums (POS) form — Hinglish notes

3.1.10 · Hardware › Boolean Algebra & Logic Gates

POS kya hai?

Maxterm kaise banate hain (first principles se)

HUM KYA CHAHTE HAIN: ek aisa single OR-term jo $0$ ho exactly ek row par.

COMPLEMENT RULE KYUN: OR tab $0$ hota hai sirf jab andar ka har literal $0$ ho. Literal $A$ tab $0$ hota hai jab $A = 0$ ; literal $\overset{ˉ}{A}$ tab $0$ hota hai jab $A = 1$ . Toh kisi chosen row par term ko $0$ banane ke liye, har variable ko complement karo agar woh variable us row mein $1$ hai.

Worked example — poori derivation

Function: $F (A, B, C) = 1$ siway rows $0, 2, 5$ ke.

#	A B C	F
0	0 0 0	0
1	0 0 1	1
2	0 1 0	0
3	0 1 1	1
4	1 0 0	1
5	1 0 1	0
6	1 1 0	1
7	1 1 1	1

Step 1 — $F = 0$ wali rows chuno: rows $0, 2, 5$ . Kyun? POS zeros se banta hai.

Step 2 — har maxterm likho (variable ko complement karo agar woh us row mein $1$ hai):

Row 0 ( $000$ ): sab zero → $(A + B + C)$ .
Row 2 ( $010$ ): $B = 1$ → $(A + \overset{ˉ}{B} + C)$ .
Row 5 ( $101$ ): $A = 1, C = 1$ → $(\overset{ˉ}{A} + B + \overset{ˉ}{C})$ .

Step 3 — AND karo: $F = (A + B + C) (A + \overset{ˉ}{B} + C) (\overset{ˉ}{A} + B + \overset{ˉ}{C})$ AND kyun? Har factor exactly apni bad row ko "kill" karta hai (output $0$ karta hai); AND karne se koi bhi ek $0$ dominant ho jaata hai.

Shorthand notation

Common mistakes

Recall Feynman: 12-saal ke bachhe ko explain karo

Socho ek light hai jo almost hamesha ON rehni chahiye, lekin tumhe pata hai 3 "forbidden" button-combos hain jo ise OFF karni chahiye. Har bure combo ke liye tum ek tripwire lagaate ho jo sirf exactly us combo par OFF hota hai. Sab tripwires ko series mein wire karo (yeh hai AND): agar koi bhi tripwire trip ho, light off ho jaati hai. Toh light sirf un 3 bure combos ke liye OFF hoti hai aur baaki sab ke liye ON. Woh series-of-tripwires hi Product of Sums hai!

Active recall

A POS (product of sums) form kis tarah ke terms ka AND hota hai?

Sum terms (literals ke ORs); canonically, maxterms.

A maxterm

M_{i}

kis value ke barabar hota hai, aur kitni rows par?

Equals

0

at exactly one row (row

i

) and

1

at all other rows.

To build canonical POS, which truth-table rows do you use?

The rows where

F = 0

In a maxterm, you complement a variable when its value in that row is?

1 (one). Plain literal when it is 0.

Why AND the maxterms together?

Because AND outputs 0 if any term is 0, so each maxterm forces 0 at its one bad row while others stay 1.

Derive POS from SOP of

\overset{ˉ}{F}

: what identity is used?

F = \overline{\overset{ˉ}{F}} = \overline{\sum m_{i}} = \prod \overline{m_{i}} = \prod M_{i}

(De Morgan;

\overline{m_{i}} = M_{i}

Maxterm for row

A = 0, B = 1, C = 0

is?

(A + \overset{ˉ}{B} + C)

Notation

\prod M (0, 2, 5)

means?

Product of maxterms for rows 0, 2, 5 (the rows where

F = 0

F = \sum m (1, 3, 4, 6, 7)

for 3 variables, its POS is?

\prod M (0, 2, 5)

(the complementary row set).

Connections

Sum of products (SOP) form — dual; $F = 1$ rows use karta hai.
Minterms and maxterms — SOP/POS ke peeche ke atoms.
De Morgan's theorems — minterm sums ko maxterm products mein convert karne ka engine.
Truth tables — dono forms ka source.
Karnaugh maps — $0$ s ko group karna simplified POS deta hai.
Logic gates — POS directly ek AND-of-ORs (two-level) circuit mein map hota hai.