The statement "The square of any rational number is rational" can be rewritten formally as "For all rational numbers x, x 2 is rational" or as "For all x, if x is rational then x 2 is rational." Rewrite each of the following statements in the two forms "∀ x, " and "∀x, if, then " or in the two forms "∀ x and y, " and "∀x and y, if, then."

a. The reciprocal of any nonzero fraction is a fraction.

b. The derivative of any polynomial function is a polynomial function.

c. The sum of the angles of any triangle is 180◦.

d. The negative of any irrational number is irrational.

e. The sum of any two even integers is even.

f. The product of any two fractions is a fraction.

a.

For all non zero fraction 1 (1/x), 1/x is a fraction

For all 1 / (1/x), if 1 / (1/x) is non zero, 1/x is a fraction

b.

For all polynomial function f (x) = x³ + x² + x - 1, the derivative (dy/dx) is a polynomial function

For all f (x) x³ + x² + x - 1, if f (x) is polynomial, f' (x) is a polynomial function

c.

For all angles x, y, z of a triangle, the sum, x + y + z = 180

For all x, y, z, if x, y and z are the angles of a triangle, x+y+z = 180

d.

For all irrational numbers x, - x is irrational

For all x, if x is irrational then - x irrational.

e.

For two integers, x and y, the sum x+y is an integer

For x, y if x and y are integers, then x + y is an integer

f.

For two fractions, x/y and a/b the product ax/by is a fraction

For x/y and a/b, if x/y and a/b are fractions then ax/by is a fraction