Laws of Boolean Algebra — Digital Logic -

🧠 Boolean Algebra & Minimization Techniques (Laws of Boolean Algebra)

When you start learning about Boolean Algebra, it might feel like diving into a world made only of 1s and 0s.
But don’t worry — this world is actually very logical (pun intended!).

Boolean Algebra helps us simplify logical expressions, which is super important in designing digital circuits — like the ones inside your computer, calculator, or even your washing machine.

To make this possible, we use a set of laws — simple rules that tell us how logic operations behave.
These laws are like grammar rules in a language — they help us write and simplify logic statements in a clear, correct way.

Let’s go through them together in plain English.

⚙️ 1. Identity Law

This one’s easy.
In Boolean Algebra, 1 means “true” and 0 means “false.”

Now, if you AND anything with 1, it stays the same.
If you OR anything with 0, it also stays the same.

In symbols:

M · 1 = M
M + 0 = M

🧩 Example:
If a light works only when a switch (A) is ON and the main power (1) is ON, the light depends only on A — because 1 doesn’t change anything.

It’s like multiplying something by 1 — it doesn’t change its value.

⚡ 2. Null Law

This one is about the “dead ends” of logic.

M + 1 = 1 → If any condition is true, the result is always true.
M · 0 = 0 → If any condition is false, the result is always false.

🧩 Example:
If a fan works only when both switches are ON, and one switch is permanently OFF (0), the fan will never run.
On the other hand, if you use an OR condition and one input is permanently ON (1), the output is always ON.

🔁 3. Idempotent Law

Doing the same thing twice doesn’t change the result.

M + M = M
M · M = M

🧩 Example:
If you already pressed the ON button, pressing it again doesn’t make it “more ON.”
The state is already ON — no need to repeat!

🔄 4. Complement Law

Every value has an opposite — called its complement.
If A = 1, then A’ (read as “A not”) = 0, and vice versa.

The laws are:

M + M’ = 1 → Either M or its opposite must be true.
M · M’ = 0 → M and its opposite can never be true together.

🧩 Example:
You can’t be awake (M) and asleep (M’) at the same time.
But you must be one of the two — awake or asleep.

🔁 5. Involution Law

Applying NOT twice brings you back to the original.

(M’)’ = M

🧩 Example:
If you say, “It’s not true that it’s not raining,” what you mean is — it is raining!
Double negatives cancel out.

🧮 6. Domination Law

Sometimes, one condition overpowers everything else.

M + 1 = 1 (Because 1 dominates OR)
M · 0 = 0 (Because 0 dominates AND)

🧩 Example:
If your friend says, “I’ll go out if it’s not raining OR if I feel like it,” — if they feel like it (1), it doesn’t matter what the weather is.
That “1” dominates the decision.

🔀 7. Double Negation Law

This is similar to the Involution Law but worth repeating:
Two NOTs cancel each other.

( (M’)’ = M )
That’s the same as saying “not not M = M.”

🧩 Example:
If you say, “I’m not unhappy,” it means — you’re happy!

🧩 8. Commutative Law

The order of inputs doesn’t matter for AND and OR operations.

M + N = N + M
M · N = N · M

🧩 Example:
Saying “M OR N” is the same as “N OR M.”
If either is true, the result is true — order doesn’t change that.

🔁 9. Associative Law

When you group things, the output doesn’t change.

(M + N) + O = M + (N + O)
(M · N) · O = M · (N · O)

🧩 Example:
Whether you check (math OR science) first and then add English, or group (science OR English) first — the final result (if any subject is passed) stays the same.

🔄 10. Distributive Law

This one feels like the math you already know!

M · (N + O) = M·N + M·O
M + (N · O) = (M + N) · (M + O)

🧩 Example:
If you have to water (plants OR grass), you can water each separately — same overall result.

🔃 11. Absorption Law

This law helps in simplification by “absorbing” repeated logic.

M + (M · N) = M
M · (M + N) = M

🧩 Example:
If M is already true, it doesn’t matter whether M AND N is true — M alone is enough.

✂️ 12. De Morgan’s Laws

These are the golden rules for handling NOTs with ANDs and ORs.
They tell us how to distribute NOT properly:

(M · N)’ = M’ + N’
(M + N)’ = M’ · N’

🧩 Example:
Imagine the rule “You can’t enter if you’re not wearing a mask AND gloves.”
That’s the same as saying: “You can enter if you’re missing a mask OR gloves.”
See how the “and” flips to “or”? That’s exactly what De Morgan’s Law does.