Boolean algebra laws (commutative, associative, distributive): Foundations

This page assumes nothing. Before you touch the parent laws note, we build every single symbol it uses, one brick at a time, each one anchored to a picture. If a symbol appears in the parent, it is defined here first.

0. What is a "bit"? The only two values in the whole universe

Ordinary counting numbers go $0, 1, 2, 3, \dots$ up forever. Boolean algebra throws almost all of them away. It keeps exactly two:

WHY only two? Because real hardware is built from switches, and a switch has exactly two honest states: current flows, or it doesn't. We invented an algebra with two values precisely because that matches the physical thing we want to describe.

Figure — Boolean algebra laws (commutative, associative, distributive)

Look at the picture: the number line of ordinary maths is an endless road, but the Boolean world is just two dots. Every symbol on this page is a machine that takes one or two of these dots and returns another dot.

1. The NOT symbol $\overline{A}$ — the bit-flipper

Our first machine has one input.

The picture: imagine a light switch wired backwards. When you press it OFF, the lamp turns ON, and vice versa. The overline is that "reverse" wire.

WHY the topic needs it: the parent note writes $\overline{A}$ when it talks about the Logic Gates AND OR NOT and later the De Morgan's Theorems. Without knowing "bar means flip", those symbols are gibberish.

Recall Check yourself

What is $\overline{\overline{A}}$ (flip, then flip again)? ::: You get $A$ back — flipping twice returns the original bit.

2. The AND symbol $A \cdot B$ — the "both" machine

Now a machine with two inputs.

Here is the complete behaviour — all four possible input pairs (this is called a truth table, defined properly in §5):

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

The picture — two switches in a row (series): current only reaches the lamp if switch $A$ and switch $B$ are both closed. One gap anywhere = darkness.

3. The OR symbol $A + B$ — the "at least one" machine

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

The picture — two switches side by side (parallel): current reaches the lamp if either path is closed. You need to block both to get darkness.

4. Precedence — why $A \cdot B + A \cdot C$ isn't ambiguous

When several symbols appear together, which do we do first?

WHY we need this rule: the parent writes $A \cdot B + A \cdot C$ with no brackets. Without a precedence rule you couldn't tell if it means " $(A \cdot B) + (A \cdot C)$ " or " $A \cdot (B + A) \cdot C$ ". The rule "AND before OR" (same as "times before plus" in normal maths) makes it unambiguous: it is $(A \cdot B) + (A \cdot C)$ .

5. The truth table — the tool that PROVES everything

Every "proof" in the parent note is a truth table. So what is it, and why is it a valid proof?

WHY this counts as a proof: in ordinary algebra you can't test $a + b = b + a$ for all numbers — there are infinitely many. But Boolean inputs only take values $0$ and $1$ . With just $2, 4,$ or $8$ rows you cover literally every case that can ever occur. If two columns match on every row, the two expressions are provably equal — no case escaped.

6. Two special constants the proofs quietly use

The parent's algebraic proof secretly uses two facts. Let's earn them now.

WHY flagged here: the parent's quick proof of $A + B C = (A + B) (A + C)$ jumps from $A (1 + C + B)$ to $A$ . That jump only works because $1 + anything = 1$ . If you haven't met that fact, the proof looks like a magic trick. Now it's just $max$ .

7. Putting the symbols together — a worked parse

You have now met every symbol the parent uses: $0$ , $1$ , $\overline{A}$ , $\cdot$ , $+$ , brackets, precedence, truth tables, and the two collapse-constants. You are ready.

Prerequisite map

Equipment checklist

Recall Cover the right side and answer aloud — you are ready when all pass

The only two values in Boolean algebra ::: $0$ and $1$ — nothing else exists. What $\overline{A}$ does ::: Flips the bit: $\overline{0} = 1$ , $\overline{1} = 0$ . What $A \cdot B$ means in one word ::: "Both" — output $1$ only if both inputs are $1$ ; equals $min (A, B)$ . What $A + B$ means in one word ::: "At least one" — output $1$ if either input is $1$ ; equals $max (A, B)$ . Why $1 + 1 = 1$ here (not $2$ ) ::: $+$ means OR $= max$ ; there is no value $2$ , so $max (1, 1) = 1$ . Order of operations ::: Bar first, then AND ( $\cdot$ ), then OR ( $+$ ); brackets override. How many rows for a 3-input truth table ::: $2^{3} = 8$ rows — every variable doubles the count. Why a truth table counts as a full proof ::: Inputs are only $0/1$ , so finitely many rows cover every possible case. Value of $A \cdot A$ ::: $A$ (idempotent), since $min (A, A) = A$ . Value of $1 + X$ for any $X$ ::: $1$ , since $max (1, X) = 1$ .

Connections

Parent topic (Hinglish) — the laws these symbols unlock.
Logic Gates AND OR NOT — the physical switches behind $\cdot$ , $+$ , $\overline{A}$ .
Boolean Algebra Identity & Null Laws — where $A \cdot A = A$ and $1 + X = 1$ live formally.
De Morgan's Theorems — the next tool that leans on the bar symbol.
Karnaugh Maps — a visual truth-table cousin.
Combinational Circuit Simplification — the payoff: fewer gates.