Question

A group G, ({0}, +) under addition operation satisfies which of the following properties?

(a) identity, multiplicity and inverse

(b) closure, associativity, inverse and identity

(c) multiplicity, associativity and closure

(d) inverse and closure

This question was addressed to me during a job interview.

My doubt is from Groups topic in section Groups of Discrete Mathematics

Answer 1

Right choice is (b) closure, associativity, inverse and identity

For explanation: Closure for all a, b∈G, the result of the operation, a+b, is also in G. Since there is one element, hence a=b=0, and a+b=0+0=0∈G. Hence, closure property is satisfied. Associative for all a, b, c∈G, (a+b)+c=a+(b+c). For example, a=b=c=0. Hence (a+b)+c=a+(b+c)

⟹(0+0)+0=0+(0+0)⟹0=0. Hence, associativity property is satisfied. Suppose for an element e∈G such that, there exists an element a∈G and so the equation e+a=a+e=a holds. Such an element is unique, the identity element property is satisfied. For example, a=e=0. Hence e+a = a+e⟹0+0=0+0⟹0=a. Hence e=0 is the identity element. For each a∈G, there exists an element b∈G (denoted as a-1), such that a+b=b+a=e, where e is the identity element. The inverse element is 0 as the addition of 0 with 0 will be 0, which is also an identity element of the structure.