Tao Analysis Vol I questions, “logically equivalence” problems

analysislogicproof-explanationproof-verification

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.

Best Answer

  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 $$X\rightarrow Y\ \&\ (Y\rightarrow Z\ \&\ Z\rightarrow X) \equiv X\rightarrow Y\ \&\ Y\rightarrow X\equiv X\leftrightarrow Y\\ (X\rightarrow Y\ \&\ Y\rightarrow Z)\ \&\ Z\rightarrow X \equiv X\rightarrow Z\ \&\ Z\rightarrow X\equiv X\leftrightarrow Z\\ Y\rightarrow Z\ \&\ (Z\rightarrow X\ \&\ X\rightarrow Y) \equiv Y\rightarrow Z\ \&\ Z\rightarrow Y\equiv Y\leftrightarrow Z $$
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