in rosen discrete mathematics book there is this example on quantifiers and logical operators
"not all primes are odd"
is equivalent in the predicate form to:
1) not for all x [p(x) implies o(x)]
2) there exists [p(x) and not o(x)]
in the second form why the author didn't use the implication is it right to write it as follows:
there exists [p(x) implies not o(x)]??

Question

in rosen discrete mathematics book there is this example on quantifiers and logical operators
"not all primes are odd"
is equivalent in the predicate form to:
1) not for all x [p(x) implies o(x)]
2) there exists [p(x) and not o(x)]
in the second form why the author didn't use the implication is it right to write it as follows:
there exists [p(x) implies not o(x)]??

anonymous · Answer

$$\(\bf\huge\color{#ff0000}{W}\color{#ff2000}{e}\color{#ff4000}{l}\color{#ff5f00}{c}\color{#ff7f00}{o}\color{#ffaa00}{m}\color{#ffd400}{e}~\color{#bfff00}{t}\color{#80ff00}{o}~\color{#00ff00}{O}\color{#00ff40}{p}\color{#00ff80}{e}\color{#00ffbf}{n}\color{#00ffff}{S}\color{#00aaff}{t}\color{#0055ff}{u}\color{#0000ff}{d}\color{#2300ff}{y}\color{#4600ff}{!}\color{#6800ff}{!}\color{#8b00ff}{!}$$

anonymous · Answer

@jwasily2

anonymous · Answer

Implication is typically a broad statement that applies to more than one thing.  In words, the second form says "There exists a number x that is prime but not odd".  You're asking if it is okay to say "There exists a number x where the fact that x is prime does not imply that it is odd".  Do you see why the second form is awkward?  Implication is not generally used with the existential qualifier because the existential qualifier is making a statement about a single element whereas implication makes a statement about a collection of elements.  

Because I'm not a discrete mathematician, I'm not sure if your form would be technically considered wrong or not, but it's certainly more awkward than the form given in the book, so I'd stick with the latter if I were you.