Question

Boolean Algebra

Prove that The OR operation is closed for all x, y ∈ B x + y ∈ B

and

Prove that The And operation is closed for all x, y ∈ B x . y ∈ B

Answer 1

A mathematical structure O is said to be closed under an operation +

if there are a,b belongs to O then a + b also belongs to O.

Q 1: Prove that The OR operation is closed for all x, y ∈ B x + y ∈ B

Proof: Let x and y belongs to Binary set B ={0,1}

if we add x and y we have four cases

0+0=0

0+1=1

1+0=1

1+1=1

so the addition of x+y also yields 0 or 1 that belongs to B

Hence x+y is closed under B.

Q 2: Prove that The OR operation is closed for all x, y ∈ B x . y ∈ B

Proof: Let x and y belongs to Binary set B ={0,1}

if we multiply x and y we have four cases

0.0=0

0.1=0

1.0=0

1.1=1

so the multiplication of x and y also yields 0 or 1 that belongs to B

Hence x.y is closed under B.