Here are the questions
My solution:
(a) Any integer has an inverse
(b) Any nature number is either even or odd
(c) There exists an identity nature number. (I don't know what it calls, 1 time any number is the number itself.)
(d) Don't know
(e) Don't know
(f) There exists a quadratic function in R (I think I should use $0$ somewhere)
Can anyone help me with (d), (e) and check the others?
a) Right idea, but you need to say here that the 'inverse' is relative to addition. That is, for every integer x there is an integer '-x' (where x + (-x) = 0)
b) Good!
c) Again, you need to be a bit more specific: this time, the 'identity' element is relative to multiplication. (Also, technically, it only says that there is a 'right-identity' element z, that is: $x*z=x$, and it does not say it is a 'left-identity' element: $z*x=x$. Typically, for something to be an identity element it needs to be a right and left identity element.)
d) This one is like a), but for multiplication: it says that every non-zero natural number has an inverse relative to multiplication, i.e. for any x there is a y such that $x*y = 1$ (which is of course not true, but that is what the statement says)
e) This one is a little weird, and I have a hard time putting it into a fluent English sentence, but basically it says that multiplication (for integers) has a left-identity element (the y), and that two times that identity element is a perfect square (which again is not true, since y would have to be 1, and 2 is not a perfect square)
f) As pointed out, this is saying that every quadratic function has a root (which is again not true, but that is what it says)