Assume that x and y are real numbers and n is a positive int

Assume that x and y are real numbers and n is a positive integer. Determine whether each of the following is true or false, and explain your conclusion.

a) For all x there exists a y with x + y = 0.

b) There exists y so that for all x, x + y = 0.

c) For all x there exists y for which xy = 0. d) There exists y so that for all x, xy = 0.

e) For all x there exists y such that xy = 1.

f) There exists y so that for all x, xy = 1.

g) For all n, n is even or n is odd.

h) Either n is even for all n, or n is odd for all n.

Solution

a) true for y=-x, it will be true. x is real so -x will be real . y=-x will real. thus there exist real y satisfying the equation.

b) true. same as above

c) true, y=0 satisfy the eqaution and 0 is real. so there exist a real y satidfying the equation.

d)true, same as above

e) false, if x=0, y does not exist

f)false, same as above

g) n is positive integer, therefore it can be either odd or even. so true

h) false , n can be even or odd.