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.