Lesson 2.2 — NOT Distributes

When NOT sits in front of a parenthesized AND or OR, it moves inside and flips the connective.

The Setup

When you put a NOT in front of an AND or an OR group, something interesting happens. The NOT goes inside and flips the connective.

This is one of the most useful tricks in logic. Mathematicians call it De Morgan's Law. We'll learn it in English first; the symbols come in Module 6.

NOT in Front of AND

"It is NOT (raining AND cold)."

This means: it's not the case that both things are happening. But that's the same as saying: at least one of them is NOT happening.

"It is NOT raining, OR it is NOT cold."

(Or both — could be neither raining nor cold, that still fits.)

The NOT goes inside, and the AND flips to OR.

NOT in Front of OR

"It is NOT (raining OR snowing)."

This means: it's not the case that either thing is happening. But that's the same as saying: both must NOT be happening.

"It is NOT raining, AND it is NOT snowing."

The NOT goes inside, and the OR flips to AND.

See It in the Truth Table

Two expressions are equivalent when they give the same truth value in every row of the truth table. The De Morgan rewrites match the originals in every row — that's the whole point. Toggle the two variables below and watch.

Look at the right two columns in either table: every row has the same value in both columns. The original form and the De Morgan rewrite are saying the same thing.

Why This Matters

Sometimes the negative version is easier to think about than the original.

"Not all the kids are happy."

Means: at least one kid is NOT happy. (That's De Morgan applied to a quantifier — Module 8 territory, but it's the same idea.)

"Not all the food is gone."

Means: at least one piece of food is left. Easier to picture than "it is not the case that all the food is gone."

The trick lets you flip a hard-to-think-about statement into an easier one. Especially useful when programming, writing legal language, or arguing carefully about anything with multiple conditions.

Tricky Bits

1. The parens matter.

NOT raining AND cold is NOT the same as NOT (raining AND cold).

The first one — without parens — means "(NOT raining) AND cold" because NOT binds tightest. That says: it's not raining, but it is cold. The second one is the De Morgan setup: "neither both raining nor cold" — at least one of them isn't happening.

Write the parens. Always.

2. De Morgan is reversible.

The law works both ways. If you see (NOT X) OR (NOT Y), you can fold it back into NOT (X AND Y). Sometimes the folded version is shorter to write or easier to read.

Practice

Rewrite each using De Morgan. Then sanity-check the rewrite with a truth-table mental walk.

1. NOT (I'm hungry AND I'm tired)
2. NOT (raining OR cold)
3. NOT (I cleaned my room AND I did my homework)
4. NOT (pizza OR pasta)
5. (NOT A) AND (NOT B)   (fold this one back into the NOT-outside form)

Answers:

1. (NOT hungry) OR (NOT tired) — at least one isn't the case.
2. (NOT raining) AND (NOT cold) — both not the case.
3. (NOT cleaned room) OR (NOT did homework) — at least one chore wasn't done.
4. (NOT pizza) AND (NOT pasta) — neither.
5. NOT (A OR B) — folded back.

Recap

• NOT (X AND Y) = (NOT X) OR (NOT Y) — NOT goes inside, AND flips to OR.
• NOT (X OR Y) = (NOT X) AND (NOT Y) — NOT goes inside, OR flips to AND.
• The law works both ways. You can fold from inside-out, or expand from outside-in.
• The two forms are equivalent — same truth value in every row.
• The De Morgan flip is one of the most-used moves in logic. Worth memorizing.