ISI2012-PCB-A-2a

Question

Professor Hijibiji has defined the following Boolean algebra $\mathcal{B} = (B, +, *)$, where

• $B = \{1, 2, 3, 5, 6, 10, 15, 30\}$, i.e., the set of all eight factors of $30$;
• the two binary operators $’+’$ and $’*’$ respectively denote the LCM (least common multiple) and GCD (greatest common divisor) of two integer operands.

Show that the two operations of $\mathcal{B}$ satisfy

1. associativity
2. commutativity
3. distributivity.

Answer 1

It satisfies all three properties:

Associative : (A+B)+C = A+(B+C)

Let's take A = 2, B =3 , C =5. These all are from set B.

Here + means LCM and * means GCD. So,

LCM(LCM(2,3),5) = 30

Same as LCM(2,LCM(3,5)) i.e. 30. So plus operation is Associative.

GCD(GCD(2,3),5) = GCD(2,GCD(3,5)) = 1 . So Mul operation is Associative.

Commutative : A + B = B + A

LCM(2,3) = LCM(3,2) = 6 and GCD(2,3) = GCD(3,2) =1 So commutative too.

Now lets talk about Distributive,

we need to find A*(B+C) = A*B + A*C

GCD(2,LCM(3,5)) = LCM(GCD(2,3),GCD(2,5)) = 1 So Set B is distributive also.

Correct Me If I am wrong.

Comments

Though these properties get satisfied but showing ex wont prove it we need to generalize it.