De Morgan's Laws Proof and real world application. De Morgan's Laws are transformational Rules for 2 Sets 1) Complement of the Union Equals the Intersection of the Complements not (A or B) = not A and not B 2) Complement of the Intersection Equals the Union of the Complements not (A and B) = not A or not B Take 2 Sets A and B Union = A U B ← Everything in A or B Intersection = A ∩ B ← Everything in A and B U = Universal Set (All possible elements in your defined universe) Complement = A’ Everything not in A, but in the Universal Set

1. DeMorgan’s Laws
Transformational Rules for 2 Sets
1. Complement of the Union Equals the Intersection of the Complements
a. not (A or B) = not A and not B
2. Complement of the Intersection Equals the Union of the Complements
a. not (A and B) = not A or not B
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2. DeMorgan’s Laws Visual
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3. Set Notation Refresher
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
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4. Prove DeMorgan’s Law #1
Complement of the Union Equals the Intersection of the Complements
Let P = (A U B)' and Q = A' ∩ B'
Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)'
⇒ x ∉ (A U B)
⇒ x ∉ A and x ∉ B
⇒ x ∈ A' and x ∈ B'
⇒ x ∈ A' ∩ B'
⇒ x ∈ Q
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5. Test Problem for DeMorgan’s Law #1
Complement of the Union Equals the Intersection of the Complements
Take 2 sets A and B. (A U B)’ = A’ ∩ B’
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}, B = {2, 4, 6}
A U B = {1, 2, 3, 4, 5, 6}, (A U B)’ = {7, 8, 9, 10}
A’ = {6, 7, 8, 9, 10}, B’ = {1, 3, 5, 7, 8, 9, 10}
A’ ∩ B’ = {7, 8, 9, 10}
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6. Prove DeMorgan’s Law #2
Complement of the Intersection Equals the Union of the Complements
Let P = (A ∩ B)' and Q = A' U B'
Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A ∩ B)'
⇒ x ∉ (A ∩ B)
⇒ x ∉ A or x ∉ B
⇒ x ∈ A' or x ∈ B'
⇒ x ∈ A' U B'
⇒ x ∈ Q
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7. Test Problem for DeMorgan’s Law #2
Complement of the Intersection Equals the Union of the Complements
Take 2 sets A and B. (A ∩ B)’ = A’ U B’
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}, B = {2, 4, 6}
A ∩ B = {2, 4}, (A ∩ B)’ = {1, 3, 5, 6, 7, 8, 9, 10}
A’ = {6, 7, 8, 9, 10}, B’ = {1, 3, 5, 7, 8, 9, 10}
A’ U B’ = {1, 3, 5, 6, 7, 8, 9, 10}
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