Question

Determine the truth value of the following statement (assuming that x,y and z are real numbers): For every x there exists a y such that for all z, z is less than or equal to x+y implies that z is less than or equal to y

Answer 1

True,

CASE 1: Lets take the instance where x is negative (x<0). For the statement that z is less than or equal to x+y to be true y must be AT LEAST the quantity (z + |x|) since, we have stated that x is not zero, it follows that y >= z (TRUE)

CASE 2: x = 0. For z to be less than or equal to x+y, y must be greater than or equal to z. (TRUE)

CASE3: x>0. for z to be less than or equal to x+y, y must be at least the quantity (z-x). However, we are free to choose a y sufficiently large such that it is greater than or equal to z since we are only required to find at least one y for each instance of x that satisfies the conditions.

Answer 2

thanks a lot! very helpful

Answer 3

except I don't know how case 3 holds

Answer 4

I would say that y doesn't have to be greater than or equal to z in that scenario, though we can certainly find a y that does. Perhaps I'm confused on what the question asks. Are we asked if there exists a y such that these conditions hold? Or are we asked that given the quantifiers on the variables, that the implication must be true?