English

# 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

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