Boolean algebra — AND, OR, NOT, XOR, NAND, NOR operations

1.1.3 · Coding › How Computers Work

KAUNSI hai har operation

YE symbols ($+$ aur $\cdot$) kyun?

Kyunki ye arithmetic ki tarah behave karte hain. $A\cdot B$ multiplication jaisa kaam karta hai: $0\cdot 0=0,1\cdot 1=1$. $A+B$ addition jaisa kaam karta hai sivaaye $1+1=1$ ke (ye saturate ho jaata hai, 1 se aage overflow nahi kar sakta). Yahi analogy poori wajah hai ise algebra kyun kehte hain.

KAISE padhein har gate ko — truth tables

Truth table har possible input combination ke liye output list karta hai. $n$ inputs ke saath $2^n$ rows hoti hain.

NOT (ek input): $\neg 0=1$, $\neg 1=0$.

Formulas scratch se derive karna

XOR ka formula yaad nahi karte — tum use banate ho. "Inputs kab alag hote hain?" Ya ($A=1,B=0$) ya ($A=0,B=1$). Har English word ko algebra mein translate karo:

• "$A=1$ aur $B=0$" $\Rightarrow A\cdot \bar{B}$
• "$A=0$ aur $B=1$" $\Rightarrow \bar{A}\cdot B$
• "ya...ya" $\Rightarrow +$

$\boxed{A\oplus B=A\bar{B}+\bar{A}B}$

Ye step kyun? Truth table ki har row jahan output = 1 hota hai, ek AND-term (minterm) banta hai; hum un sab terms ko OR karte hain. Ise Sum of Products (SOP) kehte hain aur isse tum kisi bhi truth table ka formula derive kar sakte ho.

Worked examples

Flashcards

Connections

• Logic Gates and Transistors — ye operations physical circuits kaise bante hain.
• Binary Number System — 0/1 values jinpar ye gates kaam karte hain.
• Half Adder and Full Adder — XOR + AND se binary addition banti hai.
• Karnaugh Maps — Boolean expressions simplify karne ka visual tool.
• Sum of Products and Minterms — kisi bhi truth table se formula derive karna.
• De Morgan's Laws — AND aur OR ke beech ka bridge.

Concept Map