Question

Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)

Answer 1

Let’s assume,

A={ 1,2 }(1)

B={ 2,3 }

A∩B={ 2 }

A−B=A−( A∩B ) ={ 1,2 }−{ 2 } ={ 1 }

( A∩B )∪( A−B )={ 2 }∪{ 1 } ={ 1,2 } (2)

From equations (1) and (2),

A=( A∩B )∪( A−B )

Hence,

A=( A∩B )∪( A−B )

B−A=B−( A∩B ) ={ 2,3 }−{ 2 } ={ 3 }

A∪( B−A )={ 1,2 }∪{ 3 } ={ 1,2,3 } (3)

( A∪B )={ 1,2,3 }(4)

From equations (3) and (4),

A∪( B−A )=( A∪B )

Hence,

A∪( B−A ) = ( A∪B )