GATE | GATE MOCK 2017 | Question 38

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

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

(C) ( A, + ) is a group

(D) ( A, + ) is not a semi group

Answer: (B)

Explanation:
Given, A = { 1,2,3,4,…….∞ }
A: ( A,+ ) to be semi group, it has to satisfy Closure property & Associative property

Closure: Given, a + b = ab => 1 + 2 = 3
2 + 3 = 5

So what ever value we take for a,b their ab value belongs to A so it satisfies closure.
Associative: I n order to satisfy associative property it needs to hold

( a + b) + c = a + ( b + c )
Checking:

( a + b) + c = a + ( b + c )
ab + c = a + bc

abc = abc
So for all values for a,b,c associative property satisfies

Monoid:
a + e = e + a = a, ∀ a ∈ A

ae = a
e = 1

Identity element is 1, so A is monoid.
Group:

a + b = b + a = e
It doesn’t satisfy the property because for all values of a,b it is not equal to e. So it not a group.

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