What is cartesian product/concatenation of empty string with another set and empty set with another set?

Question

Let $L$ be set of strings.

What is $L\times \{\}$ and what is $L\times\epsilon$ where $\times$ is cartesian product?

What is $L\circ\{\}$ and what is $L\circ\epsilon$ where $\circ$ is concatenation?

Answer 1

$L \times \{\} =\{\}$ ( prefer just to say $\emptyset$ instead of $\{\}$, because there are no pairs with the second component in $\emptyset$.

$L \times \varepsilon$ is not defined, as $L$ is a set (of strings), and $\varepsilon$ is not (it's the empty string). Or if it is short for the language only having $\varepsilon$ in it, so $\{\varepsilon\}$ it's just the set $\{(a,\varepsilon) :a \in L\}$.

$L \circ \emptyset = \{a \circ x: a \in L, x \in \emptyset \} = \emptyset$ again, as there are no second strings to choose.

But $L \circ \varepsilon$ is just $L$ as $x = x \circ \varepsilon$ for all $x \in L$.