Foundations — Half adder and full adder

Before you can read a single equation on the Half adder and full adder parent note, you must own every symbol it throws at you. This page builds them from nothing, in an order where each one leans on the one before it. Nothing is assumed.

0. What is a "bit"? (the atom)

The picture: a single lamp. Dark or bright — no dimmer.

Why the topic needs it: every wire in an adder carries exactly one bit at a time. The whole subject is "what to do when several of these switches are on."

1. Counting in binary (why $1+1$ isn't just "2")

We usually count in decimal: ten symbols $0,1,2,\dots ,9$, and when we run past $9$ we start a new column — $10$ means "one ten, zero ones."

Binary does the same trick but with only two symbols. When you run past $1$, you start a new column.

So $10_2$ (the little $2$ means "read me in binary") is: $10_2=(1\times 2)+(0\times 1)=2\text{ in decimal.}$

The picture: two lamps both on. Their total "two" spills into a second lamp on the left.

2. The two outputs: Sum and Carry

This is identical to carrying in ordinary addition. When you add $7+5=12$ by hand, you write 2 and carry 1. Here, $1+1=10_2$, so you write $S=0$ and carry $C=1$.

Why the topic needs it: "half adder" and "full adder" are just the names of the little machines that compute $S$ and $C$.

3. Boolean values vs. Boolean operations

We now need a language for "this output is 1 only when...". That language is Boolean algebra: algebra where every variable is a bit ($0$ or $1$).

A letter like $A$ or $B$ just names a bit — a stand-in for "whatever value that wire is carrying right now." $A$ might be $0$ on Tuesday and $1$ on Wednesday; the equations must hold either way.

Three operations are all we use. Meet them one at a time.

3a. NOT — the bar $\bar{A}$

The picture: a switch wired backwards — the lamp is on exactly when you press off.

Why we need it: to say "$A$ is zero AND $B$ is one" we write $\bar{A}B$. The bar lets us test for a bit being off.

3b. AND — the dot $A\cdot B$

The picture: two switches in series on one wire — current flows only if both are closed.

Why we need it: the half-adder Carry is exactly "both bits are 1" → $C=A\cdot B$. See AND gate.

3c. OR — the plus $A+B$

The picture: two switches in parallel — current flows if either is closed.

3d. XOR — the circled plus $A\oplus B$

This is the star of the show, so it earns its own build.

Why "exclusive"? Plain OR says yes to $1+1$; XOR excludes the "both" case — it wants one or the other, not both.

The deep reason the topic loves XOR: look at the Sum column of the half-adder. $S=1$ exactly when $A$ and $B$ differ. That is XOR. And for three inputs, XOR chained ($A\oplus B\oplus C_{in}$) outputs $1$ whenever an odd number of inputs are $1$ — because each incoming $1$ flips the running answer. That "odd counter" behaviour is precisely a Sum bit. See XOR gate.

4. Truth tables and minterms (the derivation method)

Why exactly $2^n$ rows? Each input independently doubles the number of cases: 1 input → 2, 2 inputs → 4, 3 inputs → 8. Covering all of them is what guarantees the circuit never meets a case you forgot.

The method the whole parent note uses: for any output, OR together the minterms of the rows where that output is $1$. That gives a guaranteed-correct Boolean formula, which you then simplify. This is the engine behind Truth tables and minterms.

Example (half-adder Sum): $S=1$ on rows $(0,1)$ and $(1,0)$, so $S=\bar{A}B+A\bar{B}=A\oplus B.$

5. Subscripts: $C_{in}$, $C_{out}$, $S_0$, $10_2$

Subscripts are just name tags, never operations:

• $C_{in}$ = the carry coming into this column from the right.
• $C_{out}$ = the carry going out of this column to the left.
• $S_0,S_1,S_2$ = the Sum bits of column 0, 1, 2 (column 0 is rightmost, worth 1).
• The little $2$ in $10_2$ = "this numeral is written in binary."

Why the topic needs them: in real addition you chain many adders. Each column's $C_{out}$ becomes the next column's $C_{in}$ — that chaining is the Ripple-carry adder. Without subscripts you couldn't tell the carries apart.

6. "Majority function" — a name you'll meet

$\text{Majority}=AB+AC+BC$

Read it as three "pairs both on" tests OR-ed together — if any pair is both-on, at least two inputs are 1. This is exactly the full-adder Carry-out $C_{out}$. (The Carry-lookahead adder speeds this up, but the logic is the same.)

How these feed the topic

All roads lead into the parent: Half adder and full adder. This whole topic lives inside the family of Combinational circuits — circuits with no memory, where output depends only on the current inputs.

Equipment checklist

Cover the right side and check you can answer each before moving on.