Boolean Algebraic Theorems — Digital Logic
🧠 Boolean Algebra & Minimization Techniques (Boolean Algebraic Theorems)
Let’s start with a simple thought — imagine you’re trying to simplify a long logical condition in a computer program.
Wouldn’t it be great if there were shortcuts or simple rules that tell you how to make those expressions smaller and easier to handle?
That’s exactly what Boolean algebraic theorems do in digital logic!
Boolean algebra gives us a set of rules to simplify logic equations that use variables like K, L, and M, where each variable can only have two values — 1 (True) or 0 (False).
These theorems help reduce complicated logic circuits into smaller, faster, and more efficient designs.
Let’s explore these theorems one by one, in simple, everyday language.
⚙️ 1. Commutative Theorem
This one says: The order doesn’t matter when you use AND or OR.
- K + L = L + K
- K · L = L · K
🧩 Think of it like this:
If I say, “You’ll pass if you study or practice,” it’s the same as saying “You’ll pass if you practice or study.”
The meaning stays the same.
So, the order of variables doesn’t affect the result.
🔄 2. Associative Theorem
This theorem tells us that grouping doesn’t change the result when combining more than two variables.
- (K + L) + M = K + (L + M)
- (K · L) · M = K · (L · M)
🧩 Example:
If you’re checking whether K, L, or M are online, it doesn’t matter how you group them — the final result (someone’s online or not) stays the same.
🔀 3. Distributive Theorem
This one looks similar to the distributive law in normal math.
- K · (L + M) = K·L + K·M
- K + (L · M) = (K + L) · (K + M)
🧩 Example:
If you decide to bring K along to meet either L or M, it’s like K meets both separately.
This rule helps when you need to expand or factor logic equations.
💡 4. Absorption Theorem
This theorem helps us get rid of repeated or unnecessary terms.
- K + (K · L) = K
- K · (K + L) = K
🧠 In simple terms:
If you already have K, adding “K AND L” doesn’t add any new information — K already covers that part.
🧩 Example:
If you’re already awake, saying “I’m awake and it’s morning” doesn’t change anything — you’re already awake!
🔁 5. Idempotent Theorem
Repeating the same variable doesn’t change the result.
- K + K = K
- K · K = K
🧩 Example:
Pressing the “ON” button twice won’t make the light brighter — it’s still ON.
So repeating doesn’t make a difference.
🚫 6. Involution Theorem
This one is short and simple:
Taking the complement twice brings you back to where you started.
- (K’)’ = K
🧩 Example:
If you say, “It’s not true that it’s not raining,” you’re really saying “It is raining.”
Double negatives cancel each other out.
🔄 7. De Morgan’s Theorems
These are super important — they explain how NOT (the complement) interacts with AND and OR.
- (K · L)’ = K’ + L’
→ “NOT (K AND L)” equals “(NOT K) OR (NOT L).” - (K + L)’ = K’ · L’
→ “NOT (K OR L)” equals “(NOT K) AND (NOT L).”
🧩 Example:
If the rule says “You cannot enter if you don’t have both a ticket and an ID,”
that’s the same as saying “You can enter if you’re missing either a ticket or an ID.”
Notice how AND becomes OR — that’s De Morgan’s magic!
🔃 8. Redundancy Theorem
Sometimes, part of a logic expression repeats information we already have.
This theorem helps you drop those extra pieces.
- K + K’·L = K + L
- K · (K’ + L) = K · L
🧩 Example:
If K means “You’re an adult” and L means “You have a license,”
then saying “You’re an adult OR (not an adult but have a license)”
is the same as saying “You’re an adult OR you have a license.”
Simpler, right?
🧮 9. Consensus Theorem
This theorem removes unnecessary middle terms that don’t affect the overall outcome.
- K·L + K’·M + L·M = K·L + K’·M
🧠 Think of it like this:
If two terms already cover all possibilities, the third one doesn’t change the result.
It’s just extra baggage.
🔁 10. Transposition Theorem
This theorem connects AND and OR operations with complements.
- K + L = (K’ · L’)’
- K · L = (K’ + L’)’
It’s basically a reverse form of De Morgan’s theorem and helps when switching between AND and OR expressions.