Digital Electronics — Logic Gates, Boolean Algebra, Flip-Flops & Counters — Basic Electronics & Communication Engineering Notes

Last reviewed 16 Sept 2026 · 9 min read

Analog and digital signals

Analog — continuous values (e.g. temperature sensor voltage).
Digital — discrete levels, typically binary (0 and 1) — noise immunity, easy storage and processing, reproducibility.

Decimal → base b: repeated division by b for the integer part (remainders read bottom-up); repeated multiplication by b for fractions.
Base b → decimal: sum of digit × b^position.
Binary ↔ octal: group bits in threes; binary ↔ hexadecimal: group in fours (from the binary point).

Binary addition: 0 + 0 = 0; 0 + 1 = 1; 1 + 1 = 10 (sum 0, carry 1); 1 + 1 + 1 = 11.
1's complement — invert all bits.
2's complement — 1's complement + 1; used to represent negative numbers and perform subtraction by addition.
n-bit 2's complement range: $- 2^{n - 1}$ to $2^{n - 1} - 1$ (8-bit: −128 to +127).

Code	Description
BCD (8421)	Each decimal digit coded in 4 bits (0000–1001)
Excess-3	BCD + 3 — self-complementing
Gray code	Successive values differ in only one bit — shaft encoders, reduces errors
ASCII	7-bit alphanumeric code (e.g. 'A' = 65)
Parity bit	Error detection (even/odd parity)

Binary to Gray: MSB same; each next Gray bit = XOR of adjacent binary bits.

Gate	Expression	Output is 1 when
AND	$Y = A \cdot B$	All inputs are 1
OR	$Y = A + B$	Any input is 1
NOT	$Y = \overline{A}$	Input is 0
NAND	$Y = \overline{A \cdot B}$	Not all inputs are 1
NOR	$Y = \overline{A + B}$	All inputs are 0
XOR	$Y = A \oplus B = A \overline{B} + \overline{A} B$	Inputs differ (odd number of 1s)
XNOR	$Y = \overline{A \oplus B}$	Inputs are equal

Universal gates: NAND and NOR — any logic function can be built using only NAND gates (or only NOR gates).

Law	Expressions
Identity	$A + 0 = A$ ; $A \cdot 1 = A$
Null	$A + 1 = 1$ ; $A \cdot 0 = 0$
Idempotent	$A + A = A$ ; $A \cdot A = A$
Complement	$A + \overline{A} = 1$ ; $A \cdot \overline{A} = 0$
Involution	$\overline{\overline{A}} = A$
Commutative, associative	As in ordinary algebra
Distributive	$A (B + C) = A B + A C$ ; $A + B C = (A + B) (A + C)$
Absorption	$A + A B = A$ ; $A (A + B) = A$ ; $A + \overline{A} B = A + B$
De Morgan's theorems	$\overline{A + B} = \overline{A} \cdot \overline{B}$ ; $\overline{A \cdot B} = \overline{A} + \overline{B}$
Consensus	$A B + \overline{A} C + B C = A B + \overline{A} C$

Sum of products (SOP) — OR of AND terms (minterms, where output = 1).
Product of sums (POS) — AND of OR terms (maxterms, where output = 0).
Karnaugh map (K-map) — grid in Gray-code order; group adjacent 1s in powers of two (1, 2, 4, 8), as large as possible, including wrap-around; don't-care conditions (X) may be used to enlarge groups.

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