📝 OR — Linguistics vs Math

When we use the word “or” in everyday language, we often mean “one or the other, but not both.”

This is called exclusive OR (XOR)

For example:

“You can have coffee or tea.” (Usually implies you can’t have both).
“Are you going to the park or the mall?” (You can’t go to both at the same time).

💡 The Logic of Inclusive OR

In mathematics, computer science, and logic, the term OR by itself, always means inclusive OR It means “A, or B, or both.”

Think of it this way:

A OR B is TRUE if:

A is TRUE
B is TRUE
BOTH A and B are TRUE

This is precisely why the absorption law holds true.

The expression $A \lor (A \land B)$ is true whenever $A$ is true.

The additional term $A \land B$ doesn’t add any new conditions for the overall statement to be true because if $A \land B$ is true, then A must already be true.

The absorption law is a fundamental concept in Boolean algebra, demonstrating that a compound term is redundant if one of its components is already present in a simpler form.

📚 Visualizing with Sets

The same principle applies directly to set theory, where:

Inclusive OR ($\lor$) is the Union ($\cup$)
AND ($\land$) is the Intersection ($\cap$)

The law $A \cup (A \cap B) = A$ makes perfect sense because the intersection $A \cap B$ is always fully contained within the set $A$

Taking the union of $A$ with a part of itself will always just result in $A$

Always assume “OR” is inclusive in a logical or mathematical context unless it is explicitly specified as exclusive OR. This simple rule prevents many common logical mistakes.

Last updated on 2025-11-08

📐 Rank 🔁 A ⊕ B ≡ A ⊙ B'