Logic - Reference Sheet: All Laws Used in Logical Equivalence
Reference Sheet: All Laws Used in Logical Equivalence
Across all the tautology, contradiction, and equivalence problems we’ve solved so far, you’ll notice the same handful of laws showing up again and again, just combined differently. This sheet collects every law used, in both notations side by side — Boolean form (+, ·, overline) and standard Logic form (∧, ∨, ¬).
1. Commutative Law
What it says: Order doesn’t matter for OR or AND.
| Boolean Form | Logic Form |
|---|---|
| A + B = B + A | A ∨ B ≡ B ∨ A |
| A · B = B · A | A ∧ B ≡ B ∧ A |
2. Associative Law
What it says: Grouping/bracket placement doesn’t matter for OR or AND, as long as the operation stays the same throughout.
| Boolean Form | Logic Form |
|---|---|
| (A + B) + C = A + (B + C) | (A ∨ B) ∨ C ≡ A ∨ (B ∨ C) |
| (A · B) · C = A · (B · C) | (A ∧ B) ∧ C ≡ A ∧ (B ∧ C) |
3. Idempotent Law
What it says: Repeating the same variable with itself changes nothing.
| Boolean Form | Logic Form |
|---|---|
| A + A = A | A ∨ A ≡ A |
| A · A = A | A ∧ A ≡ A |
4. Double Negation (Involution) Law
What it says: Negating something twice brings it back to the original.
| Boolean Form | Logic Form |
|---|---|
| A̿ = A | ¬¬A ≡ A |
5. Identity Law
What it says: OR-ing with False (0) or AND-ing with True (1) changes nothing.
| Boolean Form | Logic Form |
|---|---|
| A + 0 = A | A ∨ False ≡ A |
| A · 1 = A | A ∧ True ≡ A |
6. Domination (Null) Law
What it says: OR-ing with True (1) always gives True; AND-ing with False (0) always gives False — the “dominant” value wins no matter what the other term is.
| Boolean Form | Logic Form |
|---|---|
| A + 1 = 1 | A ∨ True ≡ True |
| A · 0 = 0 | A ∧ False ≡ False |
Used for: the fastest way to prove a Tautology — if you can manipulate an expression until a bare “+1” appears anywhere, you’re instantly done, the whole thing collapses to 1.
7. Complement Law
What it says: A variable OR-ed with its own negation is always True; AND-ed with its own negation is always False.
| Boolean Form | Logic Form |
|---|---|
| A + A̅ = 1 | A ∨ ¬A ≡ True |
| A · A̅ = 0 | A ∧ ¬A ≡ False |
8. Distributive Law (Two Forms — Important!)
What it says: AND distributes over OR, and (unlike ordinary arithmetic) OR also distributes over AND. This second form doesn’t exist in normal number algebra, so it always feels surprising the first time.
| Boolean Form | Logic Form |
|---|---|
| A(B + C) = AB + AC | A ∧ (B ∨ C) ≡ (A∧B) ∨ (A∧C) |
| A + BC = (A + B)(A + C) | A ∨ (B ∧ C) ≡ (A∨B) ∧ (A∨C) |
9. Absorption Law
What it says: A bigger, more complicated term gets “absorbed” into a simpler one that already contains it.
| Boolean Form | Logic Form |
|---|---|
| A + AB = A | A ∨ (A∧B) ≡ A |
| A(A + B) = A | A ∧ (A∨B) ≡ A |
10. De Morgan’s Laws
What it says: Negating an AND turns it into an OR of negations, and negating an OR turns it into an AND of negations. Works for any number of terms, not just two.
| Boolean Form | Logic Form |
|---|---|
| (A · B)‾ = A̅ + B̅ | ¬(A ∧ B) ≡ ¬A ∨ ¬B |
| (A + B)‾ = A̅ · B̅ | ¬(A ∨ B) ≡ ¬A ∧ ¬B |
| A₁·A₂···Aₙ‾‾‾‾‾‾‾ = A̅₁+A̅₂+···+A̅ₙ | ¬(A₁∧A₂∧···∧Aₙ) ≡ ¬A₁∨¬A₂∨···∨¬Aₙ |
11. Conversion Rules
| Rule | Form |
|---|---|
| Conversion of Implication | A → B ≡ ¬A ∨ B |
| Conversion of Biconditional | A ↔ B ≡ (A → B) ∧ (B → A) ≡ (¬A∨B) ∧ (¬B∨A) |
Every single problem we solve follows the same overall rhythm:
- Convert (→ and ↔ )
- De Morgan’s + Double Negation (get rid of all negations)
- Distribute / Expand (open up all brackets)
- Complement + Identity/Domination (kill the 0’s and 1’s that appear)
- Absorb / Commute / Associate (arrange the terms for a clean expression)