Q1: What is the order of the group $G = (\mathbb{R}, +)$, written as $\mathrm{ord}(G)$?

A1: $1$

A2: $2$

A3: $\infty$

A4: $n$

Q2: What is the order of the group $G = ({ e,a }, \circ) $ where $a \circ a = e$.

A1: $1$

A2: $2$

A3: $\infty$

A4: $n$

Q3: Let $(S, \circ)$ be a semigroup. What is the statement of the right-cancellation property?

A1: If $x b = y b$, then $x = y$.

A2: If $x b = y a$, then $x = y$.

A3: If $b x = y a$, then $x = y$.

A4: If $b x = b y$, then $x = y$.

Q4: Let $(S, \circ)$ be a semigroup given by the following sets of matrices together with the matrix multiplication: $$ \left{ \begin{pmatrix} a & b \ 0 & 0 \end{pmatrix} \biggm| a,b \in \mathbb{R} \right} $$ Do we have the right-cancellation property here?

A1: No, we find counterexamples!

A2: Yes!