Boolean Algebra Laws

Boolean Algebra Laws

1. 3 Operations on Sets (Set Theory)

   • Intersection ∩
     • Negation Law: A ∩ ~A = Empty Set
       • This means: All elements that are both in the set and not in the set (nothing is both in the basket and outside the basket)
     • Unit Law: Universal ∩ A = A
       • This means: What are the elements in both A and the Universe? Being that A is a finite set, it confines the result to the elements only in A...
         • IE: How to get the same element by ∩ with something (Unit) ?
     • Elements in both sets
     • Idempotent Law: A ∩ A = A
       • Remember: Idempotent means Unchanged in value following operation on itself.
         • We can safely intersect anything with itself and the set will remain the same
     • Associative Law: (A ∩ B) ∩ C = A ∩ (B ∩ C)
     • Commutative Law: A ∩ B = B ∩ A
     • Distributive Law: A ∩ (B v C) = A ∩ B v A ∩ C
     • De Morgan's: ~(A ∩ B) = ~A ∪ ~B
   • Union ∪
     • Negation Law: A ∪ ~A = Universal
       • This means: All elements in the set OR not in the set (everything)
     • Elements in at least one set (or)
     • Unit Law: Empty Set ∪ A = A
     • Commutative Law: A ∪ B = B ∪ A
       • Remember: Commutative means order of operands does not matter
         • We can change order of operands
       • Elements in at least one of A or B = Elements in at least B or A
     • Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
       • Remember: Association means the order of operations does not matter
         • We can change order of operation
     • De Morgan's: ~(A ∪ B) = ~A ∩ ~B
   • Complement ~
     • Double Complement Law: ~~A = A
   • Universal Set
     • Truth is universal
   • Empty Set

2. 3 Operations on Prepositions (Boolean Logic)

   • AND ^
     • Negation Law: P ^ ~P = F
     • Unit Law: P ^ T = P
     • Idempotent Law: P ^ P = P
     • Associative Law: (p ^ q) ^ r = p ^ (q ^ r)
     • Commutative Law: P ^ Q = P ^ Q
     • Distributive Law: P ^ (Q V R) = P ^ Q v P ^ R
       • Remember: Distribution means outer operation gets "distributed"/repeated over inner operations
         • We can "pull" repeated operation over operands
     • De Morgan's: ~(P ^ Q) = ~P v ~Q
   • OR v
     • Negation Law: P V ~P = T
     • Unit Law: P V F = P
     • Commutative Law: P v Q = P v Q
     • Associative Law: (p v q) v r = p v (q v r)
     • De Morgan's: ~(P v Q) = ~P ^ ~Q
       • Remember: De Morgan's Law says: We can distribute negation over the operands if we flip the operation (and becomes or)
         • Similarly, we can "pull" negation over operands if we flip the operation
   • Negation ~
     • Double Negation Law: ~~P = P
   • TRUE
   • FALSE
     • Lies are empty

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