Boolean Algebra Laws

gargantua
Mind Map by , created over 4 years ago

Mind Map on Boolean Algebra Laws, created by gargantua on 04/06/2015.

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gargantua
Created by gargantua over 4 years ago
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Boolean Algebra Laws
1 3 Operations on Sets (Set Theory)
1.1 Intersection ∩
1.1.1 Negation Law: A ∩ ~A = Empty Set
1.1.1.1 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)
1.1.2 Unit Law: Universal ∩ A = A
1.1.2.1 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...
1.1.2.1.1 IE: How to get the same element by ∩ with something (Unit) ?
1.1.3 Elements in both sets
1.1.4 Idempotent Law: A ∩ A = A
1.1.4.1 Remember: Idempotent means Unchanged in value following operation on itself.
1.1.4.1.1 We can safely intersect anything with itself and the set will remain the same
1.1.5 Associative Law: (A ∩ B) ∩ C = A ∩ (B ∩ C)
1.1.6 Commutative Law: A ∩ B = B ∩ A
1.1.7 Distributive Law: A ∩ (B v C) = A ∩ B v A ∩ C
1.1.8 De Morgan's: ~(A ∩ B) = ~A ∪ ~B
1.2 Union ∪
1.2.1 Negation Law: A ∪ ~A = Universal
1.2.1.1 This means: All elements in the set OR not in the set (everything)
1.2.2 Elements in at least one set (or)
1.2.3 Unit Law: Empty Set ∪ A = A
1.2.4 Commutative Law: A ∪ B = B ∪ A
1.2.4.1 Remember: Commutative means order of operands does not matter
1.2.4.1.1 We can change order of operands
1.2.4.2 Elements in at least one of A or B = Elements in at least B or A
1.2.5 Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
1.2.5.1 Remember: Association means the order of operations does not matter
1.2.5.1.1 We can change order of operation
1.2.6 De Morgan's: ~(A ∪ B) = ~A ∩ ~B
1.3 Complement ~
1.3.1 Double Complement Law: ~~A = A
1.4 Universal Set
1.4.1 Truth is universal
1.5 Empty Set
2 3 Operations on Prepositions (Boolean Logic)
2.1 AND ^
2.1.1 Negation Law: P ^ ~P = F
2.1.2 Unit Law: P ^ T = P
2.1.3 Idempotent Law: P ^ P = P
2.1.4 Associative Law: (p ^ q) ^ r = p ^ (q ^ r)
2.1.5 Commutative Law: P ^ Q = P ^ Q
2.1.6 Distributive Law: P ^ (Q V R) = P ^ Q v P ^ R
2.1.6.1 Remember: Distribution means outer operation gets "distributed"/repeated over inner operations
2.1.6.1.1 We can "pull" repeated operation over operands
2.1.7 De Morgan's: ~(P ^ Q) = ~P v ~Q
2.2 OR v
2.2.1 Negation Law: P V ~P = T
2.2.2 Unit Law: P V F = P
2.2.3 Commutative Law: P v Q = P v Q
2.2.4 Associative Law: (p v q) v r = p v (q v r)
2.2.5 De Morgan's: ~(P v Q) = ~P ^ ~Q
2.2.5.1 Remember: De Morgan's Law says: We can distribute negation over the operands if we flip the operation (and becomes or)
2.2.5.1.1 Similarly, we can "pull" negation over operands if we flip the operation
2.3 Negation ~
2.3.1 Double Negation Law: ~~P = P
2.4 TRUE
2.5 FALSE
2.5.1 Lies are empty