Now that we're about to dip our toe into the sea of logic, it's good to do an exercise involving the **Booleans**:

$$\mathbf{Bool} = \\{ \textrm{true}, \textrm{false} \\} .$$

This set becomes a poset where \$$\textrm{false}\leq\textrm{false}\$$, \$$\textrm{false}\leq\textrm{true}\$$, and \$$\textrm{true}\leq\textrm{true}\$$, but \$$\textrm{true}\not\leq\textrm{false}\$$. In other words \$$A\leq B\$$ in the poset if \$$A\$$ implies \$$B\$$, often denoted \$$A\implies B\$$.

In any poset \$$A \vee B\$$ stands for the **join** of \$$A\$$ and \$$B\$$: the least element of the poset that is greater than both \$$A\$$ and \$$B\$$. The join may not exist, but it is unique.

In the poset \$$\mathbb{B}\$$, what is

* \$$\textrm{true} \vee \textrm{false}\$$?

* \$$\textrm{false} \vee \textrm{true}\$$?

* \$$\textrm{true} \vee \textrm{true}\$$?

* \$$\textrm{false} \vee \textrm{false}\$$?