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Boolean Algebra Laws Cheat Sheet by

Boolean Algebra Laws and theorems cheat sheet

Operations with 0 and 1:

1. X + 0 = X
1D. X·1 = X
2. X + 1 = 1
2D. X·0 = 0

Idempotent Laws:

3. X + X = X
3D. X·X = X

Involution Law:

4. (X´)´ = X

Laws of Comple­men­tarity:

5. X + X´ = 1
5D. X·X´ = 0

Commut­ative Laws"

6. X + Y = Y + X
6D. XY = YX

Associ­ative Laws:

7. (X + Y) + Z = X + (Y + Z) = X + Y + Z
7D. (XY)Z = X(YZ) = XYZ

Distri­butive Laws:

8. X(Y + Z) = XY + XZ
8D. X + YZ = (X + Y)(X + Z)
 

Simpli­fic­ation Theorems:

9. XY + XY´ = X
9D. (X + Y)(X + Y´) = X
10. X + XY = X
10D. X(X + Y) = X
11. X(X´ + Y) = XY
11D. X + X´Y = X + Y
Try these first.

DeMorgan's Laws:

12. (X + Y + Z + . . .)´ = X´Y´Z´. . .
12D. (XYZ. . .)´ = X´ + Y´ + Z´ + . . .

Duality:

13. (X + Y + Z)^D = XYZ. . .
13D. (XYZ. . .)^D = X + Y + X + . . .
Almost never used.

Multip­lying Out & Factoring Theorem

14. (X + Y)(X´ + Z) = XZ + X´Y
14D. XY + X´Z = (X + Z)(X´ + Y)

Consensus Theorem:

15. XY + YZ + X´Z = XY + YZ
15D. (X + Y)(Y + Z)(X´ + Z) = (X + Y)(X´ + Z)
If Simpli­fic­ation Theorems don't work, try this.

XOR & XNAND

XOR: X @ Y = X´Y + XY´
XNAND: (X @ Y)´ X = Y = X´Y´ + XY
 

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