geek mock 2017 #38

Question

Let A = { 1,2,3,4,…….∞ } and a binary operation ‘+’ is defined by a + b = ab ∀ a,b ∈ A. Which of the following is true ?

( A, + ) is a semi group but not monoid

( A, + ) is a monoid but not group

( A, + ) is a group

( A, + ) is not a semi grou

 

i think it should be semigroup because we dont have an identity element

Answer 1

1> Closure Props

Here 1+2=2 , 2+3 =6 and so on....

Here it satisfies closure props

2> Associativity Props--

(1+2)+3 =6 =1+(2+3)

satisfied

3> Identity Props---

1+2 = 2 = 2+1

1+3 = 3= 3+1

here identity is 1

Satisfied

4> Inverse props-

here inverse element doesn't exist as inverse of 4 is 1/4 which is not in domain .

Condition 1, 2 , 3 satisfied which means it is monoid but condition 4 not satisfied there fore it is not a group .

hence B

Answer 2

Its semigroup.

Now, we check if idenity element exists.

let 'e' be identity element.

$\therefore$ a+ e = ae = a

$\therefore$ a(e-1) = 0

$\therefore$ There is no unique identity when a=0

But, a $\in$ {1, 2, 3......}

Thus, identity element e=1

If inverse were to exist, it would have been multiplicative inverse which is rational number  and not natural number.

Hence, inverse doesnt exist.

So, its a monoid and hence, option (B)

Comments

how you can say e=1 is an identity...

23+1=231 according to defintion of operation..

it's semigroup but not monoid...