Let's Do Some Abstract algebra!

Algebra Level 4

Consider the set of integers $\mathbb Z$ with an operation $*$ defined by, $a * b = a^b$ for all $a, b \in \mathbb Z$ then consider the following statements:

(1) : $\mathbb Z$ with operation $*$ is groupoid.
(2) : $\mathbb Z$ with an operation $*$ is group.
(3) : $\mathbb Z$ with an operation $*$ is not a group.

Both (1) and (3) are correct Both (1) and (2) are true (3) is true, and others are false

1 solution

Akash Patalwanshi
May 27, 2016

Relevant wiki: Group Theory

Let consider $2, -3∈\mathbb Z$ then,

$2*-3 = 2^{-3} = \frac{1}{8}∉ \mathbb Z$ Hence $\mathbb Z$ with an operation $*$ is not a groupoid.

Hence it is not a group too. Because every group is groupoid.

So $3$ is correct and other are false.

Let's Do Some Abstract algebra!

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