XOR, NAND, NOR, XNOR gates — Hinglish notes

3.1.6 · Hardware › Boolean Algebra & Logic Gates

1. Har gate ko first principles se banana

NAND — "NOT AND"

WHY ye formula? NAND ka matlab literally hai "AND ko invert karo". To AND lo ( $A \cdot B$ ), phir usse NOT karo (overbar). Bas itna hi.

A	B	A·B	NAND = $\overline{A \cdot B}$
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

NOR — "NOT OR"

WHY? OR ko invert karo. OR tab $1$ hota hai jab koi bhi input $1$ ho; usse invert karne par output tab $1$ hoga jab kuch bhi $1$ na ho.

A	B	A+B	NOR = $\overline{A + B}$
0	0	0	1
0	1	1	0
1	0	1	0
1	1	1	0

XOR — "exclusive OR"

HOW hum ye expression derive karte hain? Truth table padho aur Sum-Of-Products use karo: har us row ke liye ek AND term likho jo $1$ output kare, phir unhe OR karo.

A	B	XOR	Kaun si row?
0	0	0	—
0	1	1	$\overline{A} B$
1	0	1	$A \overline{B}$
1	1	0	—

Jo rows $1$ deti hain: $A = 0, B = 1$ → $\overline{A} B$ , aur $A = 1, B = 0$ → $A \overline{B}$ . Inhe OR karo: $A \oplus B = \overline{A} B + A \overline{B}$

XNOR — "exclusive NOR" (equivalence)

WHY $A B + \overline{A} \overline{B}$ ? Wahi SOP trick: "same" wali rows hain $A = B = 0$ (→ $\overline{A} \overline{B}$ ) aur $A = B = 1$ (→ $A B$ ).

2. Universality — WHY NAND (aur NOR) duniya par raaj karte hain

Teen basics ko NAND se derive karna (first principles se):

NOT: dono inputs ko ek saath jod do. $A NAND A = \overline{A \cdot A} = \overline{A}$ . ✅ Ye step kyun? $A \cdot A = A$ (idempotence), to overbar $\overline{A}$ deta hai.
AND: pehle NAND, phir NOT. $\overline{\overline{A \cdot B}} = A \cdot B$ . To AND = (A NAND B) ko ek NAND-as-NOT mein feed karo. ✅
OR: pehle har input ko invert karo, phir NAND karo: $\overline{\overline{A} \cdot \overline{B}} = A + B$ by De Morgan. ✅

Isliye chip industry billions of identical NAND cells etch kar sakti hai aur unhe kuch bhi banane ke liye wire kar sakti hai.

3. Worked examples

4. Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bachche ko explain karo

Do light switches imagine karo.

XOR: hallway ki light ON hai sirf tab jab dono switches disagree karein — ek upar, ek neeche.
XNOR: light ON hai sirf tab jab wo agree karein.
NAND: ek aalsi alarm jo chup rehta hai sirf tab jab dono buttons ek saath dabaaye jaayein.
NOR: ek choosy alarm jo beep karta hai sirf tab jab koi bhi koi button na chuaye. Aur magic trick: NAND alarm ka kaafi bada dher lekar tum har doosra gate rebuild kar sakte ho — jaise LEGO jahan ek brick se sab kuch banta hai.

5. Recall

Flashcards

NAND output 0 kab hota hai?

Sirf jab sabhi inputs 1 hon (ye AND ka inverse hai).

NOR output 1 kab hota hai?

Sirf jab sabhi inputs 0 hon (OR ka inverse).

XOR tab 1 output karta hai jab...?

Inputs alag hon.

XNOR tab 1 output karta hai jab...?

Inputs same hon (equivalence).

XOR ke liye Boolean expression?

\overline{A} B + A \overline{B}

XNOR ke liye Boolean expression?

A B + \overline{A} \overline{B}

Ek NAND se NOT kaise banate hain?

Dono inputs ko ek saath jodte hain:

A NAND A = \overline{A}

NAND ko universal kyun kehte hain?

NOT, AND, OR sabhi sirf NAND se banaaye ja sakte hain, isliye koi bhi circuit ban sakta hai.

NOR ke liye De Morgan?

\overline{A + B} = \overline{A} \cdot \overline{B}

Bits 1,0,1 ka XOR (chained) kitna hai?

0 — 1s ki even sankhya.

OR aur XOR mein farq?

Ye sirf (1,1) par alag hain: OR=1, XOR=0.

NAND se OR kaise banate hain?

Har input ko invert karo phir NAND karo:

\overline{\overline{A} \cdot \overline{B}} = A + B

Connections

Boolean Algebra Laws — De Morgan's theorem in saari conversions ko power karta hai
AND, OR, NOT gates — wo basics jinse ye bante hain
Sum of Products (SOP) — XOR/XNOR expressions kaise derive hoti hain
Half Adder and Full Adder — XOR = sum bit, AND = carry
Parity Bits & Error Detection — chained XOR parity check karta hai
Karnaugh Maps — gate expressions simplify karna
Universal Gates — NAND/NOR completeness