Suppose you know that whenever $X$ is true, then $Y$ is true; that whenever $Y$ is true, then $Z$ is true; and whenever $Z$ is true, then $X$ 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.
We are given that whenever $X$ is true, then $Y$ is true. This is $X\implies Y$.
We are given that whenever $Y$ is true, then $Z$ is true. This is $Y\implies Z$.
We are given that whenever $Z$ is true, then $X$ is true. This is $Z\implies X$.
To show $X\iff Y$ we need to show both $X\implies Y$ and $Y\implies X$. We are given $X\implies Y$. If $Y$ is true, then by $Y\implies Z$, we know $Z$ is true. Since $Z$ is true, then by $Z\implies X$, we know $X$ is true. From $Y$ we have shown $X$, that is $Y\implies X$. So we have shown $X\iff Y$.
To show $Y\iff Z$ we need to show both $Y\implies Z$ and $Z\implies Y$. We are given $Y\implies Z$. If $Z$ is true, then by $Z\implies X$, we know $X$ is true. Since $X$ is true, then by $X\implies Y$, we know $Y$ is true. From $Z$ we have shown $Y$, that is $Z\implies Y$. So we have shown $Y\iff Z$.
To show $X\iff Z$ we need to show both $X\implies Z$ and $Z\implies X$. We are given $Z\implies X$. If $X$ is true, then by $X\implies Y$, we know $Y$ is true. Since $Y$ is true, then by $Y\implies Z$, we know $Z$ is true. From $X$ we have shown $Z$, that is $X\implies Z$. So we have shown $Y\iff Z$.
By showing all three $X\iff Y$, $Y\iff Z$, and $X\iff Z$, we have shown $X,Y,Z$ are all logically equivalent.
