Algebra Commutative Property of Set Theory Proof

Commutative law is used to change the order of the operands without changing the end result.

Algebra Commutative Property

Statement:
 
First Law :
First law states that the union of two sets is the same no matter what the order is in the equation.
A ∪ B = B ∪ A
 
Proof : A ∪ B = B ∪ A
Consider the first law, A ∪ B = B ∪ A
Let x ∈ A ∪ B.
If x ∈ A ∪ B then x ∈ A or x ∈ B
x ∈ A or x ∈ B
x ∈ B or x ∈ A [according to definition of union]
x ∈ B ∪ A
x ∈ A ∪ B => x ∈ B ∪ A
Therefore,
A ∪ B ⊂ B ∪ A --- 1
 
Consider the first law in reverse, B ∪ A = A ∪ B
Let x ∈ B ∪ A.
If x ∈ B ∪ A then x ∈ B or x ∈ A
x ∈ B or x ∈ A
x ∈ A or x ∈ B [according to definition of union]
x ∈ A ∪ B
x ∈ B ∪ A => x ∈ A ∪ B
Therefore,
B ∪ A ⊂ A ∪ B --- 2
From equation 1 and 2 we can prove
A ∪ B = B ∪ A
 
Second Law :
Second law states that the intersection of two sets is the same no matter what the order is in the equation.
A ∩ B = B ∩ A
 
Proof : A ∩ B = B ∩ A
Consider the second law, A ∩ B = B ∩ A
Let x ∈ A ∩ B.
If x ∈ A ∩ B then x ∈ A and x ∈ B
x ∈ A and x ∈ B
x ∈ B and x ∈ A [according to definition of intersection]
x ∈ B ∩ A
x ∈ A ∩ B => x ∈ B ∩ A
Therefore,
A ∩ B ⊂ B ∩ A --- 3
 
Consider the second law in reverse, B ∩ A = A ∩ B
Let x ∈ B ∩ A.
If x ∈ B ∩ A then x ∈ B and x ∈ A
x ∈ B and x ∈ A
x ∈ A and x ∈ B [according to definition of intersection]
x ∈ A ∩ B
x ∈ B ∩ A => x ∈ A ∩ B
Therefore,
B ∩ A ⊂ A ∩ B --- 4
From equation 3 and 4 we can prove the Commutative Property
A ∩ B = B ∩ A
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