Boolean Algebra Laws for UIL Computer Science

Identity Laws

• $X + 0 = X$
  • Explanation: Adding $0$ changes nothing.
• $X \cdot 1 = X$
  • Explanation: Multiplying by $1$ changes nothing.

Null (Domination) Laws

• $X + 1 = 1$
  • Explanation: OR with $1$ always becomes $1$.
• $X \cdot 0 = 0$
  • Explanation: AND with $0$ always becomes $0$.

Idempotent Laws

• $X + X = X$
  • Explanation: ORing something with itself is still itself.
• $X \cdot X = X$
  • Explanation: ANDing something with itself is still itself.

Complement Laws

• $X + X' = 1$
  • Explanation: A value OR its opposite is always true.
• $X \cdot X' = 0$
  • Explanation: A value AND its opposite can never both be true.

Involution Law

• $(X')' = X$
  • Explanation: Double negation goes back to the original.

Commutative Laws

• $X + Y = Y + X$
  • Explanation: OR order doesn’t matter.
• $X \cdot Y = Y \cdot X$
  • Explanation: AND order doesn’t matter.

Associative Laws

• $X + (Y + Z) = (X + Y) + Z$
  • Explanation: Grouping ORs doesn’t change meaning.
• $X \cdot (Y \cdot Z) = (X \cdot Y) \cdot Z$
  • Explanation: Grouping ANDs doesn’t change meaning.

Distributive Laws

• $X \cdot (Y + Z) = (X \cdot Y) + (X \cdot Z)$
  • Explanation: AND distributes over OR.
• $X + (Y \cdot Z) = (X + Y) \cdot (X + Z)$
  • Explanation: OR distributes over AND.

Absorption Laws

• $X + (X \cdot Y) = X$
  • Explanation: OR absorbs a more complex version of itself.
• $X \cdot (X + Y) = X$
  • Explanation: AND absorbs a more complex version of itself.

De Morgan’s Laws

• $(X + Y)' = X' \cdot Y'$
  • Explanation: The opposite of OR becomes AND of opposites.
• $(X \cdot Y)' = X' + Y'$
  • Explanation: The opposite of AND becomes OR of opposites.