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Let $p_1, p_2$ and $p_3$ be three statements.

Suppose now we know that if $p_1$ is true, then $p_2$ and $p_3$ are equivalent. That is, if $p_1$ and $p_2$ are true, then $p_3$ is true, and if $p_1$ and $p_3$ are true, then $p_2$ is true.

Now I want to know if the reverse is true. That is, if $p_2$ and $p_3$ are equivalent, must $p_1$ be true?

One way to disprove this is to find a counter example. How shall I formulate the requirements that a counterexample must satisfy? I have trouble understanding how to satisfy the requirement that $p_2$ and $p_3$ imply each other, although I know it means either both $p_2$ and $p_3$ are true, or neither is true. Is an example where $p_1$ is false, $p_2$ and $p_3$ are true a counterexample?

Note that $p_1, p_2$ and $p_3$, more accurately $p_1(x), p_2(x)$ and $p_3(x)$ , are given statements about some changeable object $x$, and can't be changed except the object $x$ they are talking about. For example, in Does $X ⊥ Y \leftrightarrow X ⊥ Y | Z$ implies $(X,Y) ⊥ Z$?, to construct an example, we have to choose the three random variables $X$, $Y$ and $Z$, but can't change the three statements about them.

Thanks.

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Tim
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    Replace "$p_2\leftrightarrow p_3$" by just $q$. You are now asking if $p\to q$ implies $q\to p$. – vadim123 Sep 01 '15 at 01:14
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    what is to look for when searching for a counterexample? – Tim Sep 01 '15 at 01:18
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    @Tim: HINT: Remember that $p\to q$ is always true if $p$ is false. – Brian M. Scott Sep 01 '15 at 01:21
  • @Brian: Thanks. Is this a counterexample: $p_2$ and $p_3$ both are true, but $p_1$ isn't? How shall we find a counterexample that satisfies the requirement that $p_2$ and $p_3$ imply each other, although I know it means either both $p_2$ and $p_3$ are true, or neither is true. – Tim Sep 01 '15 at 02:26
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    @Tim: Yes, that would be a counterexample. A very simple way is to take $p_2$ and $p_3$ to be the same statement. – Brian M. Scott Sep 01 '15 at 02:29
  • @Brian: what do you mean by " take p2 and p3 to be the same statement"? p1, p2 and p3 are supposed to be given statements which can't be changed. – Tim Sep 01 '15 at 02:34
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    @Tim: You’re constructing a counterexample, so you get to choose $p_1,p_2$, and $p_3$. To construct the counterexample, let $p_1$ be any false statement, and let $p_2$ and $p_3$ be the same true statement. Then $p_2$ and $p_3$ are certainly equivalent, and neither implies $p_1$. – Brian M. Scott Sep 01 '15 at 02:46
  • @Brian: I should have added that $p_1, p_2$ and $p_3$ are given statements, which can't be changed. For an example, http://math.stackexchange.com/questions/1416468/does-x-%E2%8A%A5-y-rightleftarrow-x-%E2%8A%A5-y-z-implies-x-y-%E2%8A%A5-z – Tim Sep 01 '15 at 02:48
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    @Tim: If they’re given statements, then your question makes no sense: either $p_1$ is true, or it’s not true, and whether this has anything to do with the truth of $p_2$ and $p_3$ depends entirely on the specific statements involved. – Brian M. Scott Sep 01 '15 at 02:50
  • @Brian. In the linked post, to construct an example, we have to choose what the three random variables $X$ $Y$ and $Z$ are, though the three statements about the random variables are given and not changeable. – Tim Sep 01 '15 at 02:53
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    @Tim: In other words, you changed the statements. They aren’t specific statements until $X,Y$, and $Z$ are specified. – Brian M. Scott Sep 01 '15 at 02:54

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\begin{align} p_1 & = \Big(\ell \in \{ a,b,c,d,e,f,g \}\Big) \\ p_2 & = \Big(\ell \in \{a,b,c,w,x,y,z\}\Big) \\ p_3 & = \Big(\ell \in \{a,b,c,t,u,v,w\}\Big) \end{align}

It is easy to see with the above statements that if $p_1$ is true then $p_2$ and $p_3$ are either both true or both false.

Now suppose we don't know whether $p_1$ is true but we learn that $t=x$, $u=y$, and $v=z$. Learning that suddenly makes $p_2$ and $p_3$ equivalent, but it does not mean that $p_1$ must be true.

Michael Hardy
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  • Thanks. I am not looking for a concrete example, but a general formulation of a counterexample in terms of $p_1$, $p_2$ and $p_3$. (For a concrete example, see http://math.stackexchange.com/questions/1416468/does-x-%E2%8A%A5-y-rightleftarrow-x-%E2%8A%A5-y-z-implies-x-y-%E2%8A%A5-z) – Tim Sep 01 '15 at 01:31