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}\\)?

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

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}\\)?