Logical Equivalences

What is Logical Equivalence?

Two statements are logically equivalent if they’re just different ways of saying the same thing.

They look different, but they mean the same thing.

A Simple Example

Statement A: “I’m not NOT hungry”

Statement B: “I’m hungry”

These look different, but think about it:

When A is true, B is also true
When A is false, B is also false

They always match. So they’re equivalent.

Same meaning, different words.

How to Check Equivalence

Build a truth table for both statements. If the final columns are identical, they’re equivalent.

Example: Is $\neg(p \land q)$ the same as $\neg p \lor \neg q$ ?

$p$ = “it’s cheap”
$q$ = “it’s good”

Is “not both cheap and good” the same as “not cheap, or not good”?

$p$	$q$	$p \land q$	$\neg(p \land q)$	$\neg p$	$\neg q$	$\neg p \lor \neg q$
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

The columns match. They’re equivalent.

If every row matches, they mean the same thing.

Why Does This Matter?

If two statements are equivalent, you can swap one for the other.

Sometimes one form is easier to work with or understand.

The Important Equivalences

These are shortcuts. Once you know them, you don’t need to build a truth table every time.

1. Double Negation

$\neg(\neg p) \equiv p$

$p$ = “I’m hungry”

“I’m not NOT hungry” = “I’m hungry”

Two negatives cancel out.

2. De Morgan’s Laws

There are two of them. They tell you how NOT interacts with AND/OR.

De Morgan’s Law (AND version):

$\neg(p \land q) \equiv \neg p \lor \neg q$

$p$ = “I’m in New York”
$q$ = “I’m in LA”

“I’m not in New York and LA” = “I’m not in New York, or I’m not in LA”

When you push NOT inside, AND becomes OR.

De Morgan’s Law (OR version):

$\neg(p \lor q) \equiv \neg p \land \neg q$

$p$ = “I eat meat”
$q$ = “I eat fish”

“I don’t eat meat or fish” = “I don’t eat meat and I don’t eat fish”

When you push NOT inside, OR becomes AND.

3. Implication

$p \to q \equiv \neg p \lor q$

$p$ = “he’s not at home”
$q$ = “he’s at work”

“If he’s not at home, he’s at work” = “He’s either at home or at work”

Both sentences mean the same thing - he’s in one of those two places.

4. Contrapositive

$p \to q \equiv \neg q \to \neg p$

$p$ = “I study”
$q$ = “I pass”

“If I study, I pass” = “If I didn’t pass, I didn’t study”

Flip both sides and negate both. The meaning stays the same.

5. Commutative Laws

$p \land q \equiv q \land p$

$p \lor q \equiv q \lor p$

$p$ = “I have a dog”
$q$ = “I have a cat”

“I have a dog and a cat” = “I have a cat and a dog”

Order doesn’t matter for AND and OR.

6. Associative Laws

$(p \land q) \land r \equiv p \land (q \land r)$

$(p \lor q) \lor r \equiv p \lor (q \lor r)$

$p$ = “I’m tired”
$q$ = “I’m hungry”
$r$ = “I’m cold”

“(I’m tired and hungry) and cold” = “I’m tired and (hungry and cold)”

Grouping doesn’t matter for AND and OR.

7. Distributive Law

$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$

$p$ = “coffee”
$q$ = “cream”
$r$ = “sugar”

“Coffee with cream or sugar” = “(Coffee with cream) or (coffee with sugar)”

Either way, you’re getting coffee - just picking your add-on.

Summary

Law	Equation	What it does
Double Negation	$\neg(\neg p) \equiv p$	NOTs cancel
De Morgan’s	$\neg(p \land q) \equiv \neg p \lor \neg q$	NOT flips AND/OR
Implication	$p \to q \equiv \neg p \lor q$	“if-then” → “or”
Contrapositive	$p \to q \equiv \neg q \to \neg p$	flip + negate
Commutative	$p \land q \equiv q \land p$	order irrelevant
Associative	$(p \land q) \land r \equiv p \land (q \land r)$	grouping irrelevant
Distributive	$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$	expand AND over OR

When to Use These

You don’t need to memorize all of these right now.

If you’re ever unsure whether two statements are equivalent, just build a truth table and check if the columns match.

These laws are just shortcuts for when you get comfortable.

Key insight: Equivalent statements are interchangeable. Pick whichever form is easier to work with.