Boolean variables and operations (AND, OR, NOT)

3.1.4 · Hardware › Boolean Algebra & Logic Gates

Boolean variable kya hota hai?

KYA padh rahe hain hum: teen fundamental operations —

NOT (¬, overbar, !) — 1 input leta hai.
AND (∧, ·, &) — 2+ inputs leta hai.
OR (∨, +, |) — 2+ inputs leta hai.

SIRF TEEN KYUN? Kyunki yeh teen functionally complete hain: koi bhi logic function NOT, AND, OR sirf inhi se likha ja sakta hai. Baaki sab (NAND, XOR…) sirf inhi se bani shorthand hain.

Har operation ko meaning se derive karna (memorise nahi)

NOT — "ulta"

HOW we build its truth table — ek Boolean variable ki sirf 2 values hoti hain, toh sirf 2 rows chahiye:

$A$	$\overline{A}$
0	1
1	0

AND — "sab sach hone chahiye"

HOW we build the table — 2 inputs ⇒ $2^{2} = 4$ rows:

$A$	$B$	$A \cdot B$
0	0	0
0	1	0
1	0	0
1	1	1

OR — "kam se kam ek sach hai"

$A$	$B$	$A + B$
0	0	0
0	1	1
1	0	1
1	1	1

Figure — Boolean variables and operations (AND, OR, NOT)

Operations ka order (precedence)

Toh $\overline{A} + B \cdot C$ ka matlab hai $(\overline{A}) + (B \cdot C)$ .

Worked Examples

Worked example Example 3 —

$F = A \cdot \overline{B}$ ki poori table banao 4 rows kyunki 2 inputs hain.

$A$	$B$	$\overline{B}$	$A \cdot \overline{B}$
0	0	1	0
0	1	0	0
1	0	1	1
1	1	0	0
Row 3 kyun 1 hai: $A = 1$ aur $\overline{B} = 1$ , toh AND ke dono operands 1 hain. Yeh function matlab hai "A but not B".

Common Mistakes (Steel-manned)

Active Recall

Recall Quiz yourself (click to reveal)

Ek Boolean variable kitni values le sakta hai? → Do: 0 aur 1
$1$ aur $0$ ka AND? → 0
$1$ aur $0$ ka OR? → 1
$\overline{0}$ ? → 1
Precedence order? → NOT, phir AND, phir OR
OR plain addition kyun nahi hai? → ==kyunki $1 + 1$ ko $= 1$ rehna chahiye, yeh saturate karta hai==

Recall Feynman: ek 12-saal ke bachche ko samjhao

Light switches ki kalpana karo. NOT ek shaitaan switch hai jo hamesha woh karta hai jo tum press karte ho uska ulta. AND ek strict teacher hai: tumhe sticker tabhi milega jab tumne homework bhi kiya AND room bhi saaf kiya — ek miss kiya, koi sticker nahi. OR ek pyaari naani hai: woh tumhe ice cream degi agar garmi hai OR tum khush ho — koi bhi ek reason kaam karta hai. Bas yahi ek computer sach mein karta hai, ek second mein laakhon baar, bahut tez.

Flashcards

How many values can a Boolean variable hold?

Exactly two: 0 or 1 (false/true).

What are the three fundamental Boolean operations?

NOT (1 input), AND, OR (2+ inputs).

Truth-table result of AND when inputs are 1 and 0?

0 (AND needs all inputs = 1).

Truth-table result of OR when inputs are 1 and 0?

1 (OR needs at least one input = 1).

What is

\overline{A}

when

A = 1

0 (NOT flips the value).

Why is Boolean OR written with + but not ordinary addition?

Because it saturates: 1+1 = 1, staying within {0,1}.

Formula for AND using arithmetic on 0/1 values?

A \cdot B = A \times B

(any zero factor gives 0).

Formula for OR that keeps output in {0,1}?

A + B = A + B - A \cdot B = max (A, B)

Precedence order of Boolean operators?

NOT first, then AND, then OR.

How many rows in a truth table with n Boolean inputs?

2^{n}

rows.

Why are AND, OR, NOT special together?

They are functionally complete — every logic function can be built from them.

Correct simplification of

\overline{A + B}

\overline{A} \cdot \overline{B}

(De Morgan), NOT

\overline{A} + \overline{B}

Connections

Truth Tables — har input combination ko tabulate kaise karte hain ( $2^{n}$ rows).
Logic Gates — AND, OR, NOT ke physical circuit symbols.
De Morgan's Laws — NOT ka AND/OR ke saath interaction.
NAND and NOR gates — inhi basics se bane universal gates.
Boolean Algebra Laws — identity, null, idempotent, distributive rules.
Binary Number System — 0/1 values physically kahaan se aati hain.