Distributive Law of Set Theory Proof - Definition

Distributive Law states that, the sum and product remain the same value even when the order of the elements is altered.

First Law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Second Law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Distributive Law Property of Set Theory Proof

First Law :
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

First law states that taking the union of a set to the intersection of two other sets is the same as taking the union of the original set and both the other two sets separately, and then taking the intersection of the results.

Proof :
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Let x ∈ A ∪ (B ∩ C). If x ∈ A ∪ (B ∩ C) then x is either in A or in (B and C).
x ∈ A or x ∈ (B and C)
x ∈ A or {x ∈ B and x ∈ C}
{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}
x ∈ (A or B) and x ∈ (A or C)
x ∈ (A ∪ B) ∩ x ∈ (A ∩ C)
x ∈ (A ∪ B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) => x ∈ (A ∪ B) ∩ (A ∪ C)

Therefore,
A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)--- 1

Let x ∈ (A ∪ B) ∩ (A ∪ C). If x ∈ (A ∪ B) ∩ (A ∪ C) then x is in (A or B) and x is in (A or C).
x ∈ (A or B) and x ∈ (A or C)
{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}
x ∈ A or {x ∈ B and x ∈ C}
x ∈ A or {x ∈ (B and C)}
x ∈ A ∪ {x ∈ (B ∩ C)}
x ∈ A ∪ (B ∩ C)
x ∈ (A ∪ B) ∩ (A ∪ C) => x ∈ A ∪ (B ∩ C)

Therefore,
(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C)--- 2

From equation 1 and 2
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Second Law :
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Second law states that taking the intersection of a set to the union of two other sets is the same as taking the intersection of the original set and both the other two sets separately, and then taking the union of the results.

Proof :
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Let x ∈ A ∩ (B ∪ C). If x ∈ A ∩ (B ∪ C) then x ∈ A and x ∈ (B or C).
x ∈ A and {x ∈ B or x ∈ C}
{x ∈ A and x ∈ B} or {x ∈ A and x ∈ C}
x ∈ (A ∩ B) or x ∈ (A ∩ C)
x ∈ (A ∩ B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) => x ∈ (A ∩B) ∪ (A ∩ C)

Therefore,
A ∩ (B ∪ C) ⊂ (A ∩ B) ∪ (A ∩ C)--- 3

Let x ∈ (A ∩ B) ∪ (A ∩ C). If x ∈ (A ∩ B) ∪ (A ∩ C) then x ∈ (A ∩ B) or x ∈ (A ∩ C).
x ∈ (A and B) or (A and C)
{x ∈ A and x ∈ B} or {x ∈ A and x ∈ C}
x ∈ A and {x ∈ B or x ∈ C}
x ∈ A and x ∈ (B or C)
x ∈ A ∩ (B ∪ C)
x ∈ (A ∩ B) ∪ (A ∩ C) => x ∈ A ∩ (B ∪ C)

Therefore,
(A ∩ B) ∪ (A ∩ C) ⊂ A ∩ (B ∪ C)--- 4

From equation 3 and 4
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Hence, distributive law property of sets theory has been proved.
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