XOR, NAND, NOR, XNOR gates: Foundations — Hardware deep dive

Before you can read a single line about XOR or NAND, you need a small toolbox of symbols. This page hands you each tool, one at a time, and never uses a symbol before it is built. Read top to bottom.

0. The atom: a bit (the values $0$ and $1$ )

The picture: think of a single light switch. It is either up or down — never halfway. That is a bit. The whole of digital hardware is built from millions of these switches.

Why the topic needs it: every input and output in a logic gate is a bit. When the parent note writes a truth table full of $0$ s and $1$ s, each of those is one switch state. If "two values only" is not rock-solid, nothing else lands.

1. Inputs and outputs — the letters $A$ , $B$

The picture: a box with two wires coming in on the left (labelled $A$ and $B$ ) and one wire leaving on the right. The left wires carry the input bits; the right wire carries the answer.

Why the topic needs it: the formulas $A \cdot B$ , $A + B$ , $A \oplus B$ all describe what the box does to $A$ and $B$ . Without names for the inputs we could not write any rule down.

2. The three primitive rules: NOT, AND, OR

Everything else is built from these three. Learn them as actions on bits, not as spells.

2a. NOT — the flipper (the overbar $\overline{A}$ )

$A$	$\overline{A}$
0	1
1	0

The picture: press an inverter and the switch snaps to the opposite position.

Why the topic needs it: the "N" in NAND and NOR is this bar — NAND is "NOT of AND", written with a bar over the whole AND. XOR's formula $\overline{A} B + A \overline{B}$ is full of bars. If you cannot read $\overline{A}$ as "the opposite of $A$ ", the parent note is unreadable.

2b. AND — the "both" rule (the dot $\cdot$ )

$A$	$B$	$A \cdot B$
0	0	0
0	1	0
1	0	0
1	1	1

The picture: two switches wired in series (one after the other) on the same line. Current reaches the bulb only if both switches are closed.

Why the topic needs it: NAND is literally $\overline{A \cdot B}$ . To understand NAND you must first know what $A \cdot B$ gives on every row.

2c. OR — the "at least one" rule (the plus $+$ )

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

The picture: two switches wired in parallel (side by side, two paths to the bulb). Closing either path lights the bulb.

Why the topic needs it: NOR is $\overline{A + B}$ , and both XOR and XNOR formulas OR their terms together. The $+$ sign appears everywhere in the parent note.

3. Reading a truth table

The picture: a grid. Walk down it row by row; each row is one "what if the switches were like this?" scenario, and the last column tells you the light's state.

Why the topic needs it: every gate in the parent note is defined by its truth table. To "know a gate" simply means "be able to fill in its output column".

4. The compound symbols the topic introduces

Now the toolbox from sections 2 and 3 lets us read the new symbols without fear.

Why order matters (the bar's reach): in $\overline{A \cdot B}$ the bar sits over both letters, so you compute $A \cdot B$ first, then flip the single result. Compare $\overline{A} \cdot \overline{B}$ , where each letter is flipped first, then ANDed. These are different circuits — that is the whole point of De Morgan's theorem the parent note relies on.

0

and

1

— nothing else.

What does the overbar

\overline{A}

do to a bit?

Flips it:

0 \to 1

and

1 \to 0

When is

A \cdot B

(AND) equal to

1

Only when both

A

and

B

are

1

When is

A + B

(OR) equal to

1

When at least one of

A

B

1

What is

1 + 1

in Boolean OR?

1

— not

2

. Output is a single bit.

How many rows does a 2-input truth table have, and why?

4

, because

2 \times 2

input combinations (

2^{n}

for

n

inputs).

\overline{A \cdot B}

, which do you compute first?

The

A \cdot B

inside, then flip the result.

\overline{A + B}

the same as

\overline{A} + \overline{B}

No — De Morgan gives

\overline{A + B} = \overline{A} \cdot \overline{B}

What does

A \cdot A

equal, and why does it matter?

A

(idempotence); it's why tying both NAND inputs together gives NOT.

What does the

\oplus

symbol mean in words?

XOR — output

1

when the inputs differ.

What does the SOP recipe do?

Builds a formula by ORing one AND-term per row whose output is

1

Connections

↑ Parent topic — the gates these symbols build
AND, OR, NOT gates — the three primitives defined here in full
Boolean Algebra Laws — De Morgan and idempotence proved properly
Sum of Products (SOP) — the row-reading recipe in depth
Universal Gates — why NAND alone rebuilds all of the above
Half Adder and Full Adder — first place these gates get combined
3.1.06 XOR, NAND, NOR, XNOR gates (Hinglish) — same ideas in Hinglish

Foundations — XOR, NAND, NOR, XNOR gates

0. The atom: a bit (the values $0$ and $1$ )

1. Inputs and outputs — the letters $A$ , $B$

2. The three primitive rules: NOT, AND, OR

2a. NOT — the flipper (the overbar $\overline{A}$ )

2b. AND — the "both" rule (the dot $\cdot$ )

2c. OR — the "at least one" rule (the plus $+$ )

3. Reading a truth table

4. The compound symbols the topic introduces

5. Two facts you will lean on constantly

6. How these foundations feed the topic

Equipment checklist

Connections