edited by
448 views
2 votes
2 votes

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.

Which are the identity elements for $\mathcal{B}$?

edited by

1 Answer

2 votes
2 votes
For a Boolean Algebra with 2 operators, there must be 2 identity elements, one for each of the operators.

And as per definition, identity element is such an element that any elements operated with that element would produce the element itself i.e. a∗e=a where a is the element and e is the identity element.

So when operator is LCM , then the LCM of 1 with any of the given elements would result in that element i.e. LCM(1,30)=30 ; LCM(1,5)=5 etc for all the elements

When operator is HCF , the HCF of 30 with any other element would produce the same element i.e. HCF(2,30)=2; HCF(6,30)=6 etc for all the elements.

Hence the identity elements for the given Boolean Algebra  would be 1 and 30 .

Related questions

6 votes
6 votes
4 answers
3
1 votes
1 votes
1 answer
4
go_editor asked Jun 2, 2016
672 views
A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples.