please help me solve the question !!

Question

Let S be a nonempty subset of plane R^2 , it is known that every point (x, y) in S
satisfies “if x > 0 , then y > 0 ”. Consider the following properties possibly satisfied
by points (x, y) in S :

(I) If x ≤ 0 , then y ≤ 0 .
(II) If y ≤ 0 , then x ≤ 0 .
(III) If y > 0 , then x > 0 .

Which of the above properties will have to be satisfied by all points (x, y) in S?

 

(a) (II) only
(b) (III) only
(c) (I) and (II)
(d) (I) and (III)
(e) (II) and (III)

Answer 1

(a) II  only.

 

This follows from formal logic:  if A then B  (A —> B)   implies   not B —> not A (~B —> ~A).  

Only II has the inverse of y > 0.     You can replace A with "x > 0"  and  B with "y > 0" to see this.