What is the negation of the statement "Every odd nmber is divisible by 2".

Question

Intuitively,I think it is "no odd number is divisible by 2" or it could be "every odd number is not divisible by 2". Is this a trivial question or is there more to it? What is the correct answer? BTW it is from a text i am reading.

Answer 1

Suppose that you wanted to prove to someone that the statement no odd number is divisible by $2$ is false; what would you have to do? You’d have to show that some odd number actually is divisible by $2$. That is exactly what it means to say that the original statement is false. Thus, its negation is

there is an odd number that is divisible by $2$.

All it takes is one such counterexample to contradict the original statement; it’s not necessary that all odd numbers be divisible by $2$ in order to have that contradiction.

Added: You’re starting from every odd number is divisible by $2$, so the details are different, but the principle is the same. To show that it’s false, you’d have to show that there is an odd number that is ... what?

Answer 2

The negation will be "There is at least one odd number not divisible by 2". Note that the given statement is false assuming any counterexample, so saying all odd numbers are counterexamples is overkill.

Answer 3

"Not every odd number is divisible by $2$" or "Some odd number is not divisible by $2$". These equivalent statements, while true, are absurdly weak; and you can obviously say something much stronger; but they are literally the negation of the original (false) statement.