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Boolean Algebra Laws
- 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)
- 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) ?
- IE: How to get the same element by
∩ with something (Unit) ?
- 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...
- 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
- We can safely intersect anything
with itself and the set will remain
the same
- Remember: Idempotent means
Unchanged in value following
operation on itself.
- 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
- Negation Law: A ∩ ~A = Empty Set
- Union ∪
- Negation Law: A ∪ ~A = Universal
- This means: All
elements in the set OR
not in the set (everything)
- 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
- We can change order of operands
- Elements in at least one of A or B =
Elements in at least B or A
- Remember: Commutative means order of
operands does not matter
- Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Remember: Association means the
order of operations does not matter
- We can change order of operation
- We can change order of operation
- Remember: Association means the
order of operations does not matter
- De Morgan's: ~(A ∪ B) = ~A ∩ ~B
- Negation Law: A ∪ ~A = Universal
- Complement ~
- Double Complement Law: ~~A = A
- Double Complement Law: ~~A = A
- Universal Set
- Truth is universal
- Truth is universal
- Empty Set
- Intersection ∩
- 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
- We can "pull" repeated operation over operands
- Remember: Distribution means outer operation
gets "distributed"/repeated over inner operations
- De Morgan's: ~(P ^ Q) = ~P v ~Q
- Negation Law: P ^ ~P = F
- 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
- Similarly, we can "pull"
negation over operands if we
flip the operation
- Remember: De Morgan's
Law says: We can distribute
negation over the operands
if we flip the operation (and becomes or)
- Negation Law: P V ~P = T
- Negation ~
- Double Negation Law: ~~P = P
- Double Negation Law: ~~P = P
- TRUE
- FALSE
- Lies are empty
- Lies are empty
- AND ^
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