Let A be a set of integers closed under subtraction Prove that if A is nonempty, then 0 is in A and that if x is in A then -x is in A.

For subtraction we know that the Identity Property (or
Zero Property) holds. So 0 subtracted from any integer x gives the same
integer.

So x - 0 =
x

Subtracting x from both the sides we get x - x =
0

=> x + (-x ) =0

As
the set of integers A is closed under subtraction, if any two elements within the set
are subtracted from each other the result also is an element of
A.

So 0 has to be an element of A as x - 0 = x holds for
all integers.

Also if there is an element x in the set x +
(-x) = 0. This implies that the the presence of x necessitates the presence of -x in the
set as the set is closed for subtraction.