Boolean Algebra Laws

Interactive Guide to Understanding Logic Laws 🧠

7) De Morgan's Laws in Boolean Algebra 🔄

Law 1: NOT of OR is AND of NOTs

¬(A + B) = ¬A · ¬B

Meaning: The complement of (A OR B) equals (NOT A) AND (NOT B).

Law 2: NOT of AND is OR of NOTs

¬(A · B) = ¬A + ¬B

Meaning: The complement of (A AND B) equals (NOT A) OR (NOT B).

Truth Table Verification

A B A + B ¬(A + B) ¬A ¬B ¬A · ¬B
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0

Check: ¬(A + B) = ¬A · ¬B → matches perfectly!

A B A · B ¬(A · B) ¬A ¬B ¬A + ¬B
0 0 0 1 1 1 1
0 1 0 1 1 0 1
1 0 0 1 0 1 1
1 1 1 0 0 0 0

Check: ¬(A · B) = ¬A + ¬B → also matches perfectly!

Real-world Example:

"NOT (Rain OR Snow)" means:

It's NOT raining AND NOT snowing — both must be false to say no precipitation.


"NOT (Rain AND Snow)" means:

Either NOT raining OR NOT snowing (so if at least one is false, the whole statement is false).

Why is this useful?

  • In circuits, it allows you to replace NAND and NOR gates with combinations of NOT, AND, and OR gates.
  • Makes simplifying complex Boolean expressions easier.

Interactive Demo

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¬(A + B) = 1 | ¬A · ¬B = 1

8) Absorption Laws in Boolean Algebra 🧽

Law 1: A · (A + B) = A

A · (A + B) = A

Meaning: AND-ing A with (A OR B) just equals A.

Law 2: A + A · B = A

A + A · B = A

Meaning: OR-ing A with (A AND B) also just equals A.

This law removes redundant parts in expressions.

Truth Table Verification

A B A + B A · (A + B) Result compared to A
0 0 0 0 Same as A = 0
0 1 1 0 Same as A = 0
1 0 1 1 Same as A = 1
1 1 1 1 Same as A = 1

Check: A · (A + B) = A

A B A · B A + A · B Result compared to A
0 0 0 0 Same as A = 0
0 1 0 0 Same as A = 0
1 0 0 1 Same as A = 1
1 1 1 1 Same as A = 1

Check: A + A · B = A

Real-world Example:

Think of A as a switch controlling a lamp.

A + A · B means: the lamp is ON if A is ON OR A AND B are ON.

But if A is already ON, the second part A · B doesn't add anything new — the lamp is ON anyway.

So the expression simplifies to just A.

Summary:

Absorption laws help you simplify Boolean expressions by removing redundant terms.

This saves gates and makes circuits simpler!

Interactive Demo

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A + A·B = 0 | A = 0

9) Simplification Laws in Boolean Algebra ✨

Law 1: A + A′ = 1

A + A′ = 1

Meaning: If either A or NOT A is true, the result is always 1.

Law 2: A · A′ = 0

A · A′ = 0

Meaning: A value and its opposite can never both be true at the same time — result is always 0.

Truth Table Verification

A A′ A + A′ (OR) A · A′ (AND)
0 1 1 0
1 0 1 0

Matches both laws exactly!

Real-world Analogy:

Think of a light switch:

The switch is either ON (1) or OFF (0).

There's no third option — it can't be both ON and OFF at the same time.


ON OR OFF = 1 → One of them is always true → light can be detected as existing (binary decision).

ON AND OFF = 0 → Can't happen together → light is OFF.

Summary:

These laws reinforce the binary nature of Boolean logic — only two states: 0 or 1.

Helps in simplifying expressions and checking circuit logic.

Interactive Demo

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A + A′ = 1 | A · A′ = 0
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A
NOT
A′
OR
1
AND
0

10) Consensus Laws in Boolean Algebra ⚖️

Eliminates unnecessary terms to simplify expressions.

Law 1: AB + A′C = AB + C

AB + A′C = AB + C

Meaning: If you have two terms — one with A and another with A′ (NOT A) — then the middle term (A′C) can be absorbed into just C.

Why does it work?

  • If AB is true, we already know the output is true.
  • If AB is false, then A=0, so A′=1, and the result depends only on C.
  • So the term A′C adds nothing new beyond what C already gives.

Truth Table Verification

A B C AB A′ A′C AB + A′C AB + C
0 0 0 0 1 0 0 0
0 0 1 0 1 1 1 1
0 1 0 0 1 0 0 0
0 1 1 0 1 1 1 1
1 0 0 0 0 0 0 0
1 0 1 0 0 0 0 1
1 1 0 1 0 0 1 1
1 1 1 1 0 0 1 1

Notice: In all rows, AB + A′C = AB + C → The consensus term A′C can be eliminated.

Law 2: A + BC = A + A′B + C

A + BC = A + A′B + C

Meaning: You can add terms to make the expression simpler in logic or easier to implement.

Why this works:

  • If A is true, the expression is true no matter what.
  • If A=0, then it reduces to: 0+BC=A′B+C
  • Which simplifies further to just A′B + C.
  • This law is useful when implementing logic using only basic gates (NOT, AND, OR).

Truth Table Verification

A B C BC A + BC A′ A′B A′B + C A + A′B + C
0 0 0 0 0 1 0 0 0
0 0 1 0 1 1 0 1 1
0 1 0 0 0 1 1 1 1
0 1 1 1 1 1 1 1 1
1 0 0 0 1 0 0 0 1
1 0 1 0 1 0 0 1 1
1 1 0 0 1 0 0 0 1
1 1 1 1 1 0 0 1 1

In all rows, A + BC = A + A′B + C → The law is valid.

Summary of Consensus Laws

Law Simplifies to Meaning
AB + A′C AB + C One redundant term can be removed.
A + BC A + A′B + C Extra terms can be added for simplification or implementation benefits.

Real-world analogy:

If A is true, you don't need to check both B and C to decide the output — the decision is already made.

These laws help reduce logic complexity in circuits by removing or transforming terms.

Interactive Demo

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AB + A′C = 0 | AB + C = 0

11) Boolean Algebra Theorems 📜

Theorem: (A + B)(A + B′) = A

(A + B)(A + B′) = A

This is a valid Boolean identity. Let's verify it step by step.

Step-by-step logic:

Left side: (A + B)(A + B′)

Distribute using Boolean distributive law:

= A + AB′ + AB + BB′

But note:

  • BB′ = 0 (since B and NOT B can't both be true)
  • AB + AB′ = A(B + B′) = A(1) = A

So the full expression becomes:

= A

Valid!

Truth Table Verification

A B B′ A + B A + B′ (A + B)(A + B′) Result (A)
0 0 1 0 1 0 × 1 = 0 0
0 1 0 1 0 1 × 0 = 0 0
1 0 1 1 1 1 × 1 = 1 1
1 1 0 1 1 1 × 1 = 1 1

Left side matches the value of A in all rows.

Interactive Demo

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(A + B)(A + B′) = 0 | A = 0

4) Boolean Expressions & Equations 🔢

Boolean Expression

A Boolean expression is just a combination of:

  • Variables (like A, B, C)
  • Operators (AND, OR, NOT)
(A + B)(A + C)

This means: A OR B, ANDed with A OR C

No statement of equality — it's just a value you can calculate.

Boolean Equation

A Boolean equation is when two expressions are set equal to each other using an = sign.

AB + AC = A(B + C)

This means:

  • The expression on the left is logically the same as the one on the right
  • You can prove it using Boolean laws (like distributive law)

Using Laws to Simplify

We apply Boolean Laws (like distributive, absorption, De Morgan's, etc.) to:

  • Reduce complex expressions
  • Minimize the number of logic gates in a circuit
  • Make circuits faster, cheaper, and easier to design

Let's look at an example:

Original Expression: AB + AC

Can be simplified using distributive law:

= A(B + C)

So the Boolean Equation: AB + AC = A(B + C) is true.

Real-World Benefit:

AB + AC uses 2 AND gates + 1 OR gate (3 gates)

A(B + C) uses 1 OR gate + 1 AND gate (2 gates)

Result: Fewer gates = simpler, cheaper, faster circuit.

Summary

Term Description Example
Expression A logic formula (no = sign) (A + B)(A + C)
Equation Two expressions set equal AB + AC = A(B + C)
Why simplify? To reduce logic gates in circuits Fewer gates = simpler design