Understanding notation of sets

Question

What does it mean if you have a set suppose it is denoted $\theta = R \times (0,\infty)$. I'm a bit confused what the $\times$ represents?

Answer 1

$\times$ is the Cartesian product. Given two sets $A$ and $B$ the Cartesian product is given by $$A \times B = \{(a,b) \; | \; a \in A \text{ and } b \in B\},$$ that is, $A \times B$ is the set of all pairs of the form $(a,b)$ where $a$ is an element of $A$ and $b$ is an element of $B$.

So your specific set $\theta$ is given by $$R \times (0,\infty) = \{(a,b) \; | \; a \in R \text{ and } b \in (0, \infty)\},$$ so it is the set of all pairs $(a,b)$ where $a$ is a real number and $b$ is a positive real number.