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Suppose we want to prove the following:

If f(x) has property A then f(x) also has property B

I was told that if you prove if f(x) has property B then f(x) also has property A, this does not prove the original statement. Are there simple examples where one can see that this is the case?

jsguy
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2 Answers2

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"If $x$ is a dog then $x$ is a mammal" is provable. "If $x$ is a mammal then $x$ is a dog" is not, because of horses. You cannot interchange $A$ and $B$ and expect the two statements to be equally provable.

Randall
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You can reverse the implication, but then you have to also negate both sides, e.g. $A \implies B$ is equivalent to $¬B \implies ¬A$. This is the contrapositive, and is sometimes easier to prove than $A \implies B$ directly.

When you want to prove $A \iff B$, then you must prove both $A \implies B$ and $B \implies A$. They are not the same thing.

A simple example could be for an integer $x > 2$, if $x$ is prime, then $x$ is odd. Clearly if $x$ is odd does not imply that $x$ is prime.

However, if $x$ is not odd does imply $x$ is not prime.

BDN
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