All prime numbers are either even or odd, Is it a true statement?
$r:$All prime numbers are either even or odd, Is it a true statement?
I was studying Mathematical Logic then i came across above question. Since here connecting word is "OR" so if i separate two statement then it become
$p:$ All the prime number are even
$q:$ All the prime number are odd
Because both statement $p$ and $q$ are false so final statement $r$ must be false using truth value of statement for "OR" connective.
But my intuition says $r$ is true.
Am i thinking correct?
Please Help me in this.
$\endgroup$ 32 Answers
$\begingroup$You erroneously distributed the “all”. The correct interpretation is:
For every prime $p$: $p$ is even or $p$ is odd
Since every integer is either even or odd, we have:
($p$ is even or $p$ is odd) is true for all primes $p$
Therefore it is true — as your intuition suggested
$\endgroup$ 0 $\begingroup$$✔\quad$ All numbers are either even or odd.
$✗\quad$ Either all numbers are even, or all numbers are odd.
In general, $$∀x \;\Big(A(x)\text{ or }B(x)\Big)\quad\text{does not imply}\quad∀x A(x)\;\text{ or }\;∀x B(x);$$ however, $$∃x \;\Big(A(x)\text{ or }B(x)\Big)\quad\text{is equivalent to}\quad∃x A(x)\;\text{ or }\;∃x B(x),$$ and $$∀x \;\Big(A(x)\text{ and }B(x)\Big)\quad\text{is equivalent to}\quad∀x A(x)\;\text{ and }\;∀x B(x).$$
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