Tao Analysis Vol I questions, "logically equivalence" problems

Question

Questions from section A.1. (reading this book on my own / self-studying, not a student, this is not homework):

Edit: Apparently I had misinterpreted some or/and stuff so I am redoing my questions now:

Suppose that you have shown that whenever $X$ is true, then $Y$ is true, and whenever $X$ is false, $Y$ is false. Have you now demonstrated that $X$ and $Y$ are logically equivalent? Explain.

I believe this is saying, $(X \rightarrow Y) \land (\lnot X \rightarrow \lnot Y)$ which reduces to $(X \land Y) \lor (\lnot X \land \lnot Y)$, and this is the definition of "if and only if", so yes, they are logically equivalent.

Suppose that you have shown that whenever $X$ is true, then $Y$ is true, and whenever $Y$ is false, then $X$ is false. Have you now demonstrated that $X$ is true if and only if $Y$ is true? Explain.

$(X \rightarrow Y) \land (\lnot Y \rightarrow \lnot X)$, which reduces to $\lnot X \lor Y$, or just $X \rightarrow Y$ which is not the same as if-and-only-if.

Suppose that 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.

I think so? We know $X$ is logically equivalent to $Y$, and $Y$ is logically equivalent to $Z$, and by transitive property, $X$ is logically equivalent to $Z$, so they're all logically equivalent. I'm not sure if I am "allowed" to just invoke transitive property like that or if I have to show that it even applies here.

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.

$(X \rightarrow Y) \land (Y \rightarrow Z) \land (Z \rightarrow X)$, right? So by transitive property again, we have $(X \rightarrow Z) \land (Z \rightarrow X)$ so $X$ and $Z$ are logically equivalent, and similar for the other two combinations.

Answer 1

1. Yes, you have shown equivalence since you have $$X\rightarrow Y\ \&\ Y\rightarrow X \equiv X\leftrightarrow Y. $$
2. No. You have shown $$X\rightarrow Y\ \&\ \neg Y\rightarrow\neg X \equiv X\rightarrow Y\ \&\ X\rightarrow Y\equiv X\rightarrow Y. $$
3. Yes, since the implication operation is transitive.
4. Yes. Again, transitivity of implication. We have $$X\rightarrow Y\ \&\ Y\rightarrow Z\ \&\ Z\rightarrow X. $$ The statements $X,Y,Z$ are pairwise equivalent using the identity given at 1. since $$X\rightarrow Y\ \&\ (Y\rightarrow Z\ \&\ Z\rightarrow X) \equiv X\rightarrow Y\ \&\ Y\rightarrow X\equiv X\leftrightarrow Y\\ (X\rightarrow Y\ \&\ Y\rightarrow Z)\ \&\ Z\rightarrow X \equiv X\rightarrow Z\ \&\ Z\rightarrow X\equiv X\leftrightarrow Z\\ Y\rightarrow Z\ \&\ (Z\rightarrow X\ \&\ X\rightarrow Y) \equiv Y\rightarrow Z\ \&\ Z\rightarrow Y\equiv Y\leftrightarrow Z $$

Answer 2

Yes you have. The contrapositive of $p\implies q$ is $\lnot q\implies \lnot p$. They are equivalent.

So you have proved both directions: $X\implies Y$ and $Y\implies X$.

Secondly, no. As just pointed out, contrapositives are equivalent.

Thirdly, equivalence is transitive. So yes.

Finally, yes. This is a good way of proving several statements are equivalent...

Answer 3

So... it sounds to me that you have defined $X$ and $Y$ are "equivalent" to mean $X$ is true if and only if $Y$ is true. ANd to show that you must show two things:

i) $X\implies Y$ and ii) $Y \implies X$.

So for question 1) you are told $X \implies Y$ (so that's i); all you need to show is ii)) and the $\lnot X \implies \lnot Y$.

So you have to show that $(\lnot X \implies \lnot Y)$ implies that $Y \implies X$. This is the basic contrapositive and you can cite it as a proposition that should have be proven early on. But basically, if we assume $Y$ we can either have $X$ or $\lnot X$. If $\lnot X$ we'd have $\lnot Y$ which would make $Y$ impossible. So if $\lnot X \implies \lnot Y$ we can conclude ii) $Y \implies X$.

(So from now on. If ever you need to prove $a \implies b$ you can always choose to show $\lnot b \implies \lnot a$ instead. You have proven those statements are logically equivalent.)

For question 2) you are told $X\implies Y$ (so that is i); have to show whether ii) must be true or if it is possible for ii) to be false). ANd you are told $\lnot Y \implies \lnot X$. We need to show whether this does or does not imply ii) $Y \implies X$.

It does not. It is perfectly possible to have $Y$ true but $X$ not true. This satisfies i) $X \implies Y$ because a) $\lnot X$ means we don't ever get started on the hypothesis and b) $Y$ is not a contradictory conclusion. And this satisfies $\lnot Y \implies \lnot X$ because as $Y$, we never even get started on the hypothesis.

But this clearly violates ii) $Y \implies X$. So know those are not equvivalent.

For question 3) we know $X \iff Y$ and $Y \iff Z$ and we want to show $X \iff Z$.

So we have to prove two things i) $X \implies Z$ and ii) $Z \implies X$.

So... let's show them... on after the other.

i) Does $X \implies Z$?

Well Assume $X$. Then we know $X \iff Y$ so we know $Y$. And we know $Y \iff Z$ therefore we know $Z$. So $X \implies Z$.

So yes.

ii)Does $Z \implies X$?

Well Assume $Z$. Then we know $Y \iff Z$ so we know $Y$. And we know $X \iff Y$ therefore we know $X$. So $Z \implies X$.

So yes.

And that's it.