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2445381
Boolean Algebra Laws
Description
Mind Map on Boolean Algebra Laws, created by gargantua on 06/04/2015.
Mind Map by
gargantua
, updated more than 1 year ago
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Created by
gargantua
over 9 years ago
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Resource summary
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)
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
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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