🧠 Boolean Algebra & Minimization Techniques (Representation of Boolean Functions)
Let’s start simple.
A Boolean function is just a way of showing how certain input conditions produce an output — either 1 (true) or 0 (false).
In digital logic, we use these functions to describe how a circuit behaves.
You can think of them as the mathematical language of logic circuits.
💡 What Does a Boolean Function Really Mean?
Imagine you’re designing a light that turns on only when two switches are in certain positions.
You could describe this situation using a Boolean function.
For example:
F(J, L) = J·LThis means the light (output F) turns ON only when both J and L are ON (1).
That’s your Boolean function — a logical rule connecting inputs to an output.
🧩 Different Ways to Represent Boolean Functions
Just like a story can be told in words, pictures, or symbols —
a Boolean function can also be represented in different forms, each giving the same meaning but in a unique way.
Let’s explore them one by one 👇
1️⃣ Truth Table Representation
A truth table lists all possible combinations of inputs and shows the corresponding output for each.
Let’s take a small example with two inputs, J and L, and one output F = J + L (that’s the OR operation).
| J | L | F = J + L |
| :-: | :-: | :——-: |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Here, the table tells us exactly when the output is 1 and when it’s 0.
✅ Simple idea: The truth table is like a checklist that covers every possible input case.
2️⃣ Algebraic (Symbolic) Representation
This is the most common way we write Boolean functions — using variables and logical operators.
For example:
+means OR·(dot) means AND- A bar or apostrophe (like J′) means NOT
So,
F = J·L′ + J′·Lmeans the output is 1 when exactly one of the inputs is 1 (that’s the XOR condition).
🧠 It’s like writing a formula for your circuit — compact and easy to manipulate.
3️⃣ Canonical Forms (Standard Forms)
Sometimes, we want a Boolean function written in a special standard format — so everyone can understand it in the same way.
There are two main canonical forms:
🅐 Sum of Minterms (SOP Form)
Here, you express the function as a sum (OR) of minterms —
and each minterm represents one combination of variables where the output is 1.
For example, suppose your truth table shows F = 1 for:
- J=0, L=1
- J=1, L=0
Then,
F = J′L + JL′Each term (like J′L) is a minterm — it’s a specific input case that makes F true.
Think of it like listing all the winning cases in a game!
🅑 Product of Maxterms (POS Form)
Here, the function is written as a product (AND) of maxterms,
and each maxterm represents a case where the output is 0.
For the same example above, the POS form would be:
F = (J + L) · (J′ + L′)So, in short:
- SOP form → list of conditions where F = 1
- POS form → list of conditions where F = 0
Both say the same thing, just in different styles.
4️⃣ Graphical or Logic Diagram Representation
This is where we draw the Boolean function using logic gates.
Every term in the function becomes part of a circuit —
AND gates for multiplication (·), OR gates for addition (+), and NOT gates for inversion (′).
Example:
For F = J·L′ + N,
you’d draw:
- A NOT gate to invert L
- An AND gate for J·L′
- Then an OR gate to combine that result with N
🎯 The final output shows the same logic as your Boolean equation — but now in circuit form.
🔁 How All These Representations Are Connected
You can convert from one representation to another:
- From a truth table, you can write an SOP or POS form.
- From an algebraic expression, you can draw a logic circuit.
- And from a circuit, you can create a truth table by observing how it behaves.
Everything connects!
That’s why understanding these forms helps you move easily between theory and practical circuit design.
💬 Real-Life Analogy
Think of representing Boolean functions like describing a person in different ways:
- By name (algebraic form)
- By photo (logic diagram)
- By character traits (truth table)
Each one gives you information about the same person — just from a different angle.
Similarly, each Boolean function representation shows the same logic, but in its own format.
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