3.1.9
Koi bhi Boolean function basically un input rows ki list hoti hai jahan output 1 hota hai . SOP woh tarika hai jisse hum us list ko algebraically likhte hain: har "winning" row ke liye ek AND term banate hain jo sirf usi row par TRUE hota hai, phir un sab terms ko OR kar dete hain. "Winners ko OR karo" — bas yahi poora idea hai.
Definition Sum of ProductsEk Boolean expression SOP form mein hoti hai jab woh ek ya zyada terms ka OR ("sum") ho, jahan har term literals ka AND ("product") ho. Ek literal matlab ek variable ya to true form mein (A A ˉ F = product A B ˉ + product A ˉ C + product A B C
Sum = OR (+ Product = AND (⋅
Definition Minterm & Canonical SOPEk minterm ek aisa product term hota hai jisme har variable exactly ek baar aata hai (true ya complemented). Canonical (standard) SOP sirf minterms use karta hai — truth table ki har us row ke liye ek minterm jahan F = 1
Ek minterm ek row-detector hota hai. Variables A , B , C A = 1 , B = 0 , C = 1
Product A B ˉ C
Yeh 1 ke barabar hoga sirf tab jab A = 1 A = 1 B ˉ = 1 B = 0 C = 1
Kisi bhi doosre combination mein kam se kam ek literal 0 hoga, to AND 0 ho jayega.
To har minterm exactly ek input row par 1 output karta hai aur baaki jagah 0.
Ab un sab rows ke minterms ko OR karo jahan F = 1 koi bhi term 1 ho:
"F = 1
"F = 0
Yeh truth table ko exactly reproduce kar deta hai. Yahi proof hai ki har Boolean function ki ek SOP form hoti hai.
Method: har us row ke liye jahan output 1 ho, minterm likho (0 → complement, 1 → true), phir unhe OR karo.
Worked example Truth table se SOP
A
B
F
0
0
1
0
1
0
1
0
1
1
1
1
Step 1 — woh rows chuno jahan F = 1 rows 00, 10, 11.
Kyun? SOP sirf winners ko list karta hai.
Step 2 — har minterm likho (0→bar, 1→plain):
Row 00 → A ˉ B ˉ
Row 10 → A B ˉ
Row 11 → A B Kyun? Har product sirf apni row par TRUE hona chahiye.
Step 3 — OR karo:
F = A ˉ B ˉ + A B ˉ + A B Step 4 — (optional) simplify karo:
A ˉ B ˉ + A B ˉ = B ˉ ( A ˉ + A ) = B ˉ F = B ˉ + A B = B ˉ + A Kyun? A ˉ + A = 1 B ˉ + A B = B ˉ + A
Worked example Directly expression seF = A ( B + C ˉ ) Step 1 — AND ko OR par distribute karo: F = A B + A C ˉ Kyun? Distribution ek product-of-sums shape ko sum-of-products shape mein badal deta hai.
Ho gaya — yeh do product terms ke saath (non-canonical) SOP hai.
Worked example Canonical form mein expand karnaF = A + B ˉ A , B A = A ( B + B ˉ ) = A B + A B ˉ Kyun? Missing variable introduce karne ke liye ( B + B ˉ ) 1
B ˉ = ( A + A ˉ ) B ˉ = A B ˉ + A ˉ B ˉ A B ˉ F = A B + A B ˉ + A ˉ B ˉ = ∑ m ( 0 , 2 , 3 )
Common mistake "SOP matlab literal ADD aur MULTIPLY karna hai numbers ko."Kyun sahi lagta hai: + ⋅ 1 + 1 should be 2 Fix: Boolean algebra mein + ⋅ 1 + 1 = 1 2
Common mistake Galat bit par complement lagana.Kyun sahi lagta hai: yaad rehta hai "kahi bar lagana hai."
Fix: Rule fixed hai: us row mein variable 0 complemented likho; variable 1 plain likho. Kyunki minterm ko exactly usi row par 1 equal karna chahiye.
Common mistake SOP aur POS mein confusion.Kyun sahi lagta hai: dono AND aur OR use karte hain.
Fix: SOP = OR of ANDs (1 rows list karo). POS = AND of ORs (0 rows list karo). Outer operator batata hai kaun sa hai.
Recall Reveal karne se pehle answer do
SOP mein outer operation kya hai? → OR (sum).
Minterm mein variable ko complement kab karte hain? → Jab woh us row mein 0 ho.
Minterms ko OR karne se truth table recreate kyun hoti hai? → Har minterm exactly ek 1-row par fire karta hai, OR wahan output 1 rakhta hai, baaki jagah 0.
Canonical vs simplified SOP? → Canonical mein poore minterms hote hain (sab variables); simplified mein variables drop ho sakte hain.
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek light hai jo kuch button combos par ON honi chahiye. Har ON combo ke liye tum ek chota sa rule banate ho: "yeh button dabao AND woh button nahi dabao AND…" — ek aisa rule jo sirf apne exact combo ke liye "1!" chillata hai. Phir tum kehte ho: light ON hai agar rule 1 OR rule 2 OR rule 3 chillaye. Tumne bas saare winning combos list kiye aur unhe OR se jod diya. Yahi poora SOP form hai.
SOP = "Some cases OR-ed, each a Product."
Aur bars ke liye: "Zero gets a hat" (0 → complemented X ˉ
SOP stands for and means Sum of Products — ek OR of AND terms (sum = OR, product = AND).
SOP mein outer/joining operator hota hai OR (+
Ek minterm hota hai ek aisa product term jisme har variable exactly ek baar hota hai, true ya complemented.
Minterm mein variable complement karne ka rule complement karo agar variable us row mein 0 ho, plain rakho agar 1 ho.
Minterms ko OR karne se truth table reproduce kyun hoti hai har minterm exactly ek input row par 1 hota hai aur baaki jagah 0; OR un rows par 1 rakhta hai, baaki par 0.
Canonical SOP vs simplified SOP canonical mein poore minterms hote hain (sab variables present); simplified mein Boolean algebra se variables remove ho sakte hain.
F = A(B + C̄) ko SOP mein convert karo distribute karo: AB + AC̄.
SOP aur POS mein difference SOP = OR of ANDs (output-1 rows list karo); POS = AND of ORs (output-0 rows list karo).
Boolean algebra mein 1 + 1 equals 1 (OR), 2 nahi.
Kisi missing variable wale term ko canonical banane ki trick (X + X̄) = 1 se multiply karo taaki missing variable X introduce ho sake.
