Is R a group under multiplication?

The Mathematician — Fri, 14 Apr 2023 13:00:08 +0000

Is $\mathbb{R}$ a group under multiplication?

Proof. We want to show that

(R, \times)

is not a group under multiplication. Let

r \in \mathbb{R}

.

Associativity is easily shown by the definition of

\mathbb{R}

. Further, we know that the identity element of

\mathbb{R}

1

since

r \times 1 = 1 \times r = r

.

Lastly, assuming by contradiction, we want to show that each element of

\mathbb{R}

has an inverse. All elements of

\mathbb{R}

do have an inverse, but zero doesn’t. In other words, there exist no

r \in \mathbb{R}

such that:

\begin{align*} 0 \times r = 1, \end{align*}

since

0

cancels everything out. Therefore,

\mathbb{R}

not a group under multiplication

R Archives - Epsilonify