Truth tables — all 16 binary operations — Hinglish notes

1.1.4 · Coding › How Computers Work

WHAT hai ek truth table?

HOW hum unhe naam dete hain: Output column ko rows $00, 01, 10, 11$ ke liye upar-se-neeche ek 4-bit binary number ki tarah padho. E.g. AND $0, 0, 0, 1 \to 000 1_{2} = 1$ output karta hai, isliye AND "function #1" hai.

WHY $2^{2^{n}}$ operations hote hain (scratch se derivation)

Sabhi 16 operations ki complete table

Inputs ordered $A B = 00, 01, 10, 11$ . 4 output bits (upar→neeche) index banate hain.

#	Name	00	01	10	11	Expression
0	FALSE (contradiction)	0	0	0	0	$0$
1	AND	0	0	0	1	$A \cdot B$
2	A AND NOT B (inhibit)	0	0	1	0	$A \overset{ˉ}{B}$
3	A (left projection)	0	0	1	1	$A$
4	NOT A AND B	0	1	0	0	$\overset{ˉ}{A} B$
5	B (right projection)	0	1	0	1	$B$
6	XOR	0	1	1	0	$A \oplus B$
7	OR	0	1	1	1	$A + B$
8	NOR	1	0	0	0	$\overline{A + B}$
9	XNOR (equivalence)	1	0	0	1	$\overline{A \oplus B}$
10	NOT B	1	0	1	0	$\overset{ˉ}{B}$
11	B → A (if B then A)	1	0	1	1	$A + \overset{ˉ}{B}$
12	NOT A	1	1	0	0	$\overset{ˉ}{A}$
13	A → B (implication)	1	1	0	1	$\overset{ˉ}{A} + B$
14	NAND	1	1	1	0	$\overline{A \cdot B}$
15	TRUE (tautology)	1	1	1	1	$1$

Figure — Truth tables — all 16 binary operations

WHY NAND akela sab par raaj karta hai (functional completeness)

Worked examples

Worked example Example 2 — Implication

$A \to B = \overset{ˉ}{A} + B$ verify karo Implication tabhi false hoti hai jab premise true ho lekin conclusion false ho ( $A = 1, B = 0$ ).

$A$	$B$	$\overset{ˉ}{A}$	$\overset{ˉ}{A} + B$
0	0	1	1
0	1	1	1
1	0	0	0
1	1	0	1
Output $1, 1, 0, 1$ → function #13. Yeh match kyun karta hai? Sirf row $A = 1, B = 0$ 0 deta hai, exactly jaise required hai. ✓

Common mistakes

Flashcards

Kitne distinct 2-input binary logic operations hote hain, aur kyun?

2^{(2^{2})} = 16

; 4-row output column mein

2^{4}

possible fillings hoti hain.

n

-input Boolean functions ki sankhya ka general formula?

2^{(2^{n})}

XOR aur OR ko kaun si row distinguish karti hai?

11

row: OR

= 1

, XOR

= 0

Rows 00,01,10,11 ke liye AND ka output column?

0, 0, 0, 1

(function #1).

Ek single NAND gate se NOT kaise banate hain?

NAND (A, A) = \overset{ˉ}{A}

De Morgan use karke NAND se OR kaise banate hain?

A + B = NAND (\overset{ˉ}{A}, \overset{ˉ}{B})

Koi gate "functionally complete" kab hota hai?

Jab sirf usi gate se har Boolean function banaya ja sake (e.g. NAND, NOR).

Output column

1, 0, 0, 1

kaun sa function hai?

XNOR / equivalence (#9), tab true jab

A = B

Function #

k

aur #

(15 - k)

ka aapas mein kya rishta hai?

Yeh complements hain (saare output bits flip ho jaate hain).

Implication

A \to B

ki truth table — yeh 0 kab hoti hai?

Sirf jab

A = 1, B = 0

Recall Feynman: 12-saal ke bache ko samjhao

Socho do light switches, $A$ aur $B$ , aur ek bulb hai. Bulb ya toh ON hai ya OFF. Switches set karne ke sirf 4 tarike hain: dono off, ek off ek on (do tarike), dono on. Un 4 settings mein se har ek ke liye tum bulb ke liye ON ya OFF decide karte ho. Kyunki 4 settings mein se har ek apna yes/no choice hai, bulb ke liye $2 \times 2 \times 2 \times 2 = 16$ possible "rule books" hain. Har rule book ka ek naam hota hai jaise AND ("sirf tab on jab dono switches on hain") ya OR ("tab on jab kam se kam ek on hai"). Yahi hain saari 16 logic operations — kabhi 17vi nahi ho sakti.

Connections

Boolean algebra — symbolic rules ( $A + B$ , $A \cdot B$ , $\overset{ˉ}{A}$ ) jo in tables ke peeche hain
Logic gates — physical components jo har function implement karte hain
De Morgan's laws — kyun $\overline{A + B} = \overset{ˉ}{A} \overset{ˉ}{B}$ , NAND/NOR universality ki key
Binary numbers — kyun $00, 01, 10, 11$ aur $2^{n}$ counting
Functional completeness — NAND/NOR as universal gates
Adders and the ALU — XOR + AND milke binary addition karte hain