Exercise A.1.5
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.
Let's develop the truth table for this statement, which we'll call S.
| X | Y | Z | X $\iff$ Y | Y $\iff$ Z | S |
|---|---|---|---|---|---|
| F | F |
F | T | T | T |
| F | F | T |
T |
F | F |
| F | T | F | F | F | F |
| F | T | T | F | T | F |
| T | F | F | F | T | F |
| T |
F | T | F | F | F |
| T | T | F | T | F | F |
| T | T | T | T | T | T |
The fourth column for $X \iff Y$ represents the first condition in the question "X is true if and only if Y is true". Similarlym the fifth column for $Y \iff Z$ represents the second condition "Y is true if and only if Z is true".
The column for S is the conjunction of the previous two conditions,
$$ S = (X \iff Y) \land (Y \iff Z) $$
which we recognise as logical equivalence between all three X, Y and Z, because it is only true when all three variables have the same value.
So, yes, the conditions re sufficient to show X, Y and Z are logically equivalent. $\square$
It is interesting to note that what we have obeserved is the transitivity of the $\iff$ relation.
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