Suppose you know that $X$ is true if and only if $Y$ is true, and you know that $Y$ is true if and only if $Z$ is true. Is this enough to show that $X,Y,Z$ are all logically equivalent? Explain.
To show that $X,Y,Z$ are all logically equivalent, we need to show logical equivalence between all three variables, $X\iff Y$, $Y\iff Z$, $X\iff Z$.
The first $X\iff Y$ is given to us as $X$ is true if and only if $Y$ is true.
The second $Y\iff Z$ is given to us as $Y$ is true if and only if $Z$ is true.
The third $X\iff Z$ is not given to us. To show $X\iff Z$ we need to show both $X\implies Z$ and $Z\implies X$.
If $X$ is true, then by $X\iff Y$ we know $Y$ is true. Since $Y$ is true, then by $Y\iff Z$ we know $Z$ is true. From $X$ we have shown $Z$, that is, $X\implies Z$.
If $Z$ is true, then by $Y\iff Z$ we know $Y$ is true. Since $Y$ is true, then by $X\iff Y$ we know $X$ is true. From $Z$ we have shown $X$, that is, $Z\implies X$.
By showing both $X\implies Z$ and $Z\implies X$, we have shown $X\iff Z$, and this completes the requirements to show $X,Y,Z$ are all logically equivalent.
