Foundations — Demultiplexers

This page assumes nothing. If the parent note used a symbol, a word, or a piece of notation, we build it here from the ground up, in an order where each idea leans only on the one before it. Read it top to bottom once and the parent note will feel obvious.

0. A wire and a bit — the absolute floor

Before "input" or "output" means anything, we need to agree what travels on a wire.

Why the topic needs it. The demultiplexer moves one bit from one wire to one of several wires. If you don't know that a wire holds exactly a 0 or a 1, phrases like "force $Y_1=0$" are meaningless. Every symbol below is ultimately a bit sitting on a wire.

1. Naming the wires: $D$, $S$, $Y$

The parent note throws three letters at you immediately. Here is what each is, as a picture.

Why the subscripts? We can't call three output wires all "$Y$" — we'd never know which is which. So we number them $Y_0,Y_1,Y_2,\dots$ starting from zero. Starting from zero is not fussiness: it makes the wire's number equal the binary address that selects it (you'll see this in §4). The little number written low and to the right, like the $0$ in $Y_0$, is called a subscript — read "$Y$-nought", "$Y$-one".

2. The overline: $\overline{S}$ means "NOT"

The parent note writes $Y_0=\overline{S}D$. The bar on top is the first real piece of logic notation.

Why the topic needs it. Output $Y_0$ must fire when the address is $0$ — i.e. when $S=0$. But a wire being off is hard to "use" directly. The overline turns "$S$ is 0" into a wire that is 1 at exactly that moment, so we can feed it into an AND (next section). The bar is how the circuit says "pick me when the select is low."

3. Multiplication as AND: the dot $\cdot$

In $Y_0=\overline{S}D$ the two symbols sitting next to each other (and the dot in $D\cdot m_k$) mean AND, not ordinary number-multiply. But here's the lovely coincidence:

Why the topic needs it. "$D$ reaches $Y_0$ only when the address says so" is precisely an AND: (this is the chosen output) AND (there is data). The dot in $Y_k=D\cdot m_k$ is that gate.

4. Binary numbers and the select code $S_1{S}_0$

The parent note says "$S_1{S}_0=10$ means decimal 2" and "$n$ selects address $2^n$ outputs". This is the heart, so we build binary from zero.

$\text{value}=S_1\times 2+S_0\times 1$

Here $2^n$ (read "2 to the power $n$") just means "2 multiplied by itself $n$ times": $2^1=2$, $2^2=4$, $2^3=8$.

MSB / LSB. The parent warns about reading the code backwards. The MSB (most-significant bit) is the leftmost, heaviest wheel — $S_{n-1}$. The LSB (least-significant bit) is the rightmost, lightest — $S_0$. Convention on this whole topic: left is heavy. Read $S_1{S}_0=10$ as "$1$ in the 2s-place, $0$ in the 1s-place" = 2, never as 1.

5. The minterm $m_k$: "am I the chosen one?"

Now we can read the parent's key symbol. $m_k$ (say "em-sub-kay") appears in $Y_k=D\cdot m_k$.

Why the topic needs it. The whole equation $Y_k=D\cdot m_k$ now reads in plain English: output $k$ carries the data $D$ only when its personal lock $m_k$ is turned by the matching address. The bar (§2), the AND (§3), the binary code (§4) all combine right here. See Minterms and SOP for the broader family this belongs to.

6. Putting the symbols back together

You can now read the parent note's central line with zero mystery:

$\underset{\text{output wire }k}{\underbrace{Y_k}}=\underset{\text{the data bit}}{\underbrace{D}}\cdot \underset{\begin{array}{c}\text{lock that opens only}\\ \text{for address }k\end{array}}{\underbrace{m_k}}$

• If the address ≠ $k$: the lock is shut, $m_k=0$, so $Y_k=D\cdot 0=0$. Empty platform.
• If the address = $k$: the lock is open, $m_k=1$, so $Y_k=D\cdot 1=D$. The data lands here.
• If $D=0$: even the chosen output is $0$ — no train arrived, chosen platform still empty.

That covers every case: wrong address, right address, and no data. Nothing in the parent note can surprise you now.

Prerequisite map

Equipment checklist

Cover the right side; say each answer aloud before revealing.

Connections

• Demultiplexers — the parent topic these foundations feed.
• Minterms and SOP — the home of the $m_k$ notation built in §5.
• Decoders — pure minterm generators; a DEMUX with $D=1$ becomes one.
• Multiplexers — mirror device; same symbols, arrows reversed.
• Memory Addressing — where binary select codes (§4) pick a row.
• Bus Systems — one data wire (§1) fanned to many destinations.