Encoders and priority encoders: Foundations — Hardware deep dive

Before you can read a single equation like $Y_{0} = I_{1} + I_{3}$ , you must be fluent in the small alphabet of symbols it is built from. This page defines every one of them, from absolute zero, in the order they build on each other. Nothing here assumes you have seen digital logic before.

1 · A wire that is ON or OFF — the bit

The picture. Think of a light switch on a wall. It is either down (dark, $0$ ) or up (lit, $1$ ). There is no "half on". Every input line $I$ and every output line $Y$ on the parent page is exactly one of these switches.

Why the topic needs it. An encoder is entirely made of these two-state wires. "One input active" simply means "exactly one of these switches is up." If you cannot picture a wire as an up/down switch, nothing else will make sense.

2 · Writing numbers with only 0 and 1 — binary

The picture. Ordinary numbers use columns worth $1, 10, 100$ . Binary uses columns worth $1, 2, 4, 8$ . To read a binary number, add up the column-values wherever there is a $1$ .

Why the topic needs it. The whole job of an encoder is to output the binary code of an input's index. When the parent says " $I_{2}$ active gives output $10$ ", it means the number $2$ written in binary. Without binary, "the outputs give the index in binary" is a meaningless sentence.

3 · Naming the wires — subscripts $I_{0}, I_{1}, \dots$ and $Y_{1} Y_{0}$

The picture. Think of numbered lockers: $I_{0}$ is locker 0, $I_{3}$ is locker 3. The name and the number are painted right on the door.

Inputs are named $I$ (for Input): $I_{0}, I_{1}, \dots, I_{2^{n} - 1}$ .
Outputs are named $Y$ : $Y_{0}$ is the least significant bit (the $1$ s column), $Y_{1}$ the next ( $2$ s column), and so on up to $Y_{n - 1}$ .

Why the topic needs it. Every equation on the parent page is written with these subscripted names. $Y_{0} = I_{1} + I_{3}$ only means something once you know $Y_{0}$ is one specific output wire and $I_{1}, I_{3}$ are two specific input wires.

4 · The power notation $2^{n}$

The picture. Each extra output wire doubles how many inputs you can label. With $1$ output wire you can name $2$ inputs; with $2$ wires, $4$ inputs; with $3$ wires, $8$ inputs. A branching tree that splits in two at every level.

Why the topic needs it. A " $2^{n}$ -to- $n$ encoder" says: $n$ output wires can name $2^{n}$ input wires. For $n = 2$ that is a $4$ -to- $2$ encoder; for $n = 3$ an $8$ -to- $3$ . The whole naming scheme rests on this doubling.

5 · The reverse question — $lo g_{2}$

The picture. $2^{n}$ goes "wires → how many inputs." $lo g_{2}$ goes backwards: "how many inputs → how many wires do I need?" It is the undo button for the power.

Why this tool and not another? The parent asks: "8 buttons — how few wires carry which one?" That is exactly " $2$ to what power is $8$ ?" — a $lo g_{2}$ question. We use $lo g_{2}$ (not division, not subtraction) because information capacity doubles per wire, and $lo g_{2}$ is the only operation that inverts doubling.

6 · Logic OR — the $+$ symbol

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

The picture. Two switches wired in parallel to one lamp: press either switch and the lamp lights. Only if both are down is the lamp dark.

Why the topic needs it. Every encoder equation is a chain of ORs: $V = I_{0} + I_{1} + I_{2} + I_{3}$ literally means "valid if input 0 OR input 1 OR input 2 OR input 3 is on." See Boolean algebra simplification for how these expressions get tidied.

7 · Logic AND — writing symbols side by side

The picture. Two switches wired in series — like two gates on a fence, both must be open for you to walk through. If either is shut, no path.

Why the topic needs it. The priority term $I_{1} \overline{I_{2}}$ means "input 1 is on AND input 2 is off." That AND is how one input gets suppressed by a higher one.

8 · Logic NOT — the overbar $\overline{I}$

The picture. An inverter: feed a switch through it and up becomes down, down becomes up. A little upside-down mirror for a single wire.

Why the topic needs it. $\overline{I_{2}}$ is the phrase " $I_{2}$ is not active." In $Y_{0} = I_{3} + I_{1} \overline{I_{2}}$ , the bar is what makes input 1 step aside whenever the higher-priority input 2 is present. Remove the bar and priority breaks.

9 · Don't-care — the $\times$ symbol

The picture. A blank in a form you were told to leave empty. It is not $0$ and not $1$ — it is "irrelevant, skip it."

Why the topic needs it. In the priority truth table, once a high input is on, the lower inputs are marked $\times$ because they change nothing — the higher one already decided the answer. And when nothing is on, the outputs $Y_{1} Y_{0}$ are $\times$ because $V = 0$ tells us to ignore them entirely. Don't-cares are also what let Karnaugh maps simplify the logic so cleanly.

10 · The truth table itself

The picture. A full menu: every combination of orders on the left, the exact dish served on the right — with nothing left unspecified.

Why the topic needs it. The parent derives the encoder equations by reading a truth table. You look down each output column and ask "which input rows make this a $1$ ?" — then OR those together. The table is the source of truth from which all equations flow.

0

/LOW/OFF and

1

/HIGH/ON.

What is

1 1_{2}

in ordinary (base-ten) numbers?

3

(one

2

plus one

1

In the name

I_{2}

, what does the

2

mean, and is

I_{2}

the second or third wire?

It is the index (position from 0);

I_{2}

is the third wire because counting starts at

I_{0}

What does

2^{3}

equal, and what does

lo g_{2} 8

equal?

2^{3} = 8

and

lo g_{2} 8 = 3

— they are inverse operations.

In Boolean logic, what is

1 + 1

1

— the

+

means OR, not arithmetic addition.

What does

\overline{I_{2}}

mean?

NOT

I_{2}

: it is

1

when

I_{2}

0

and vice versa.

What does writing

I_{1} \overline{I_{2}}

(symbols side by side) mean?

AND:

I_{1}

is on AND

I_{2}

is off.

What does

\times

mean in a truth table?

Don't-care — the value is irrelevant and may be

0

1

What is the truth table's job?

To list every input combination and the exact required output for each — the complete definition of the circuit.

What does the valid bit

V = 0

tell you about the output wires?

Ignore them — no input is active, the output code is meaningless.

Connections

Encoders and priority encoders (Hinglish) — the parent topic these foundations feed into.
Boolean algebra simplification — the rules for manipulating $+$ , AND and $\overline{A}$ .
Karnaugh maps — a visual tool that uses don't-cares ( $\times$ ) to simplify logic.
Decoders — the inverse device; same alphabet of symbols.
Combinational circuits — parent chapter; memoryless logic built from these gates.

Foundations — Encoders and priority encoders

1 · A wire that is ON or OFF — the bit

2 · Writing numbers with only 0 and 1 — binary

3 · Naming the wires — subscripts $I_{0}, I_{1}, \dots$ and $Y_{1} Y_{0}$

4 · The power notation $2^{n}$

5 · The reverse question — $lo g_{2}$

6 · Logic OR — the $+$ symbol

7 · Logic AND — writing symbols side by side

8 · Logic NOT — the overbar $\overline{I}$

9 · Don't-care — the $\times$ symbol

10 · The truth table itself

11 · The valid signal $V$

Prerequisite map

Equipment checklist

Connections