Question

Let S be a nonempty subset of plane 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

Answer on Question #72480 - Math - Abstract Algebra

Question 72480:

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 $x \leq 0$, then $y \leq 0$.

(II) If $y \leq 0$, then $x \leq 0$.

(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)

Solution:

(I), (III) The point $(-5, 5)$ lies in $S$. This is a counterexample.

(II) If $y \leq 0$ and $x$ is not less or equal to 0, then $y \leq 0$ and $x > 0$, then $y \leq 0$ and $y > 0$. This is a contradiction.

Answer:

(a) (II) only.

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