Boolean variables and operations (AND, OR, NOT)

What is a Boolean variable?

WHAT we are studying: three fundamental operations —

NOT (¬, overbar, !) — takes 1 input.
AND (∧, ·, &) — takes 2+ inputs.
OR (∨, +, |) — takes 2+ inputs.

WHY only three? Because these three are functionally complete: any logic function whatsoever can be written using only NOT, AND, OR. Everything else (NAND, XOR…) is just shorthand built from these.

Deriving each operation from meaning (not memorising)

NOT — "the opposite"

HOW we build its truth table — a Boolean variable has only 2 values, so we only need 2 rows:

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

AND — "all must be true"

HOW we build the table — 2 inputs ⇒ $2^2 = 4$ rows:

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

OR — "at least one is true"

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

Figure — Boolean variables and operations (AND, OR, NOT)

Order of operations (precedence)

So $\overline{A} + B \cdot C$ means $(\overline{A}) + (B \cdot C)$ .

Worked Examples

Worked example Example 3 — Build the full table for

$F = A \cdot \overline{B}$ 4 rows because 2 inputs.

$A$	$B$	$\overline{B}$	$A\cdot\overline{B}$
0	0	1	0
0	1	0	0
1	0	1	1
1	1	0	0
Why row 3 is 1: $A=1$ and $\overline{B}=1$ , so both operands of AND are 1. This function means "A but not B".

Common Mistakes (Steel-manned)

Active Recall

Recall Quiz yourself (click to reveal)

How many values can a Boolean variable take? → Two: 0 and 1
AND of $1$ and $0$ ? → 0
OR of $1$ and $0$ ? → 1
$\overline{0}$ ? → 1
Precedence order? → NOT, then AND, then OR
Why is OR not plain addition? → ==because $1+1$ must stay $=1$ , it saturates==

Recall Feynman: explain to a 12-year-old

Imagine light switches. NOT is a naughty switch that always does the opposite of what you press. AND is a strict teacher: you get a sticker only if you did homework AND cleaned your room — miss one, no sticker. OR is a kind grandma: she gives you ice cream if it's hot OR if you're happy — either reason works. That's all a computer really does, millions of times a second, super fast.

Flashcards

How many values can a Boolean variable hold?

Exactly two: 0 or 1 (false/true).

What are the three fundamental Boolean operations?

NOT (1 input), AND, OR (2+ inputs).

Truth-table result of AND when inputs are 1 and 0?

0 (AND needs all inputs = 1).

Truth-table result of OR when inputs are 1 and 0?

1 (OR needs at least one input = 1).

What is

\overline{A}

when

A=1

0 (NOT flips the value).

Why is Boolean OR written with + but not ordinary addition?

Because it saturates: 1+1 = 1, staying within {0,1}.

Formula for AND using arithmetic on 0/1 values?

A\cdot B = A \times B

(any zero factor gives 0).

Formula for OR that keeps output in {0,1}?

A+B = A + B - A\cdot B = \max(A,B)

Precedence order of Boolean operators?

NOT first, then AND, then OR.

How many rows in a truth table with n Boolean inputs?

2^n

rows.

Why are AND, OR, NOT special together?

They are functionally complete — every logic function can be built from them.

Correct simplification of

\overline{A+B}

\overline{A}\cdot\overline{B}

(De Morgan), NOT

\overline{A}+\overline{B}

Connections

Truth Tables — how we tabulate every input combination ( $2^n$ rows).
Logic Gates — the physical circuit symbols for AND, OR, NOT.
De Morgan's Laws — how NOT interacts with AND/OR.
NAND and NOR gates — universal gates built from these basics.
Boolean Algebra Laws — identity, null, idempotent, distributive rules.
Binary Number System — where the 0/1 values physically come from.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Boolean algebra ka core idea simple hai: yahan har variable sirf do value le sakta hai — 0 ya 1, matlab OFF ya ON. Computer ke andar ek wire par ya to voltage hai ya nahi, beech ka kuch nahi hota. Isliye humein aisa maths chahiye jo sirf "haan/na" samjhe, aur wahi Boolean algebra hai.

Teen basic operations yaad rakho. NOT matlab ulta kar do — 1 ko 0, 0 ko 1. AND bahut strict hai: output tabhi 1 milega jab saare inputs 1 hon (jaise ATM: card bhi ho AND PIN bhi sahi ho). OR thoda generous hai: koi ek bhi input 1 ho to output 1 (jaise chhata chahiye agar baarish OR barf ho). AND ko multiply (.) aur OR ko plus (+) se likhte hain, par dhyaan rakho — OR mein 1+1 = 1 hota hai, 2 nahi, kyunki value {0,1} ke bahar nahi ja sakti.

Precedence bhi normal maths jaisa hi hai: pehle NOT, phir AND, phir OR. Confusion ho to brackets laga do. Sabse common galti yehi hoti hai ki log 1+1=2 likh dete hain ya NOT(A+B) ko galat todte hain — asli rule De Morgan hai: $\overline{A+B} = \overline{A}\cdot\overline{B}$ .

Yeh chhoti si cheez matter kyun karti hai? Kyunki inhi teen operations se hi processor, memory, calculator, sab kuch banta hai. Jab tumhe ye teen truth tables mooh-zabani yaad ho jaayen, to aage ke logic gates, circuits, sab aasan lagenge. 80/20 rule: bas teen tables + precedence + De Morgan pakad lo, aadha chapter clear.

Boolean variables and operations (AND, OR, NOT)

What is a Boolean variable?

Deriving each operation from meaning (not memorising)

NOT — "the opposite"

AND — "all must be true"

OR — "at least one is true"

Order of operations (precedence)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Boolean Algebra & Logic Gates

Connections