x, y, z belongs to Integers.
Option A: For all x, for all y, there exists a z such that x + y = z. Here, for each choices of x and y, we’re choosing one z such that x + y = z and we know sum of any two integer is an integer. Thus, True.
Option B: For all x, there exists a y such that x + y, equals to for all z, z. Here, for each choice of x, we’re choosing one y1 such that x + y1 = all z. Counter example – x1 + y1 = z1 is true but x1 + y1 = z2 is false. Thus, False.
Option C: There exists a x, for all y, there exists a z such that x + y = z. Here, for an integer x1, for each choice of y, we’re choosing one z such that x1 + y = z and we know sum of any two integer is an integer. Thus, True.
Option D: There exists a z, for all x, there exists a y such that x + y = z. Here, for an integer z1, for each choice of x, we’re choosing one y such that z1 – x = y and we know difference of any two integer is an integer. Thus, True.
Answer :- A, C, D
