Logical Implications Quiz Help These are fill in the blank questions.  I scored an 18 out of 25, but I don't see where I am making mistakes.  Please help me understand these conditional logic statements!
The answer options are:


*

*IMPLIES (-->)

*IS IMPLIED BY (<--)

*IFF (<-->, both "IMPLIES" and "IS IMPLIED BY")

*NONE OF THE ABOVE


I'm viewing these problems by saying "if P, then Q" in my head.  If that statement is true, then I mark IMPLIES.  I also check the converse: "if Q, then P."  If the converse is true, then I also mark IS IMPLIED BY.  Marking both is equivalent to IFF.  If neither "if P, then Q" or its converse are true, I mark NONE OF THE ABOVE.


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*x > 1 ________ x $\ge$ 1 Answer: NONE

*x > 1 ________ x > 0 Answer: IMPLIES

*x > 1 ________ x + 1 > 0 Answer: IMPLIES

*x > 1 ________ x - 1 > 0 Answer: IFF

*x > 1 ________ x$^2$ > 1 Answer: IMPLIES

*x > 1 ________ x + y > y Answer: IMPLIES

*x > 1 ________ xy>y Answer: IFF

*x>1 ________ x>-x Answer: IMPLIES

*x>1 ________ x>y AND y>1 Answer: IMPLIED BY

*x >1 ________ x>y OR y>1 Answer: NONE

*x>1 ________ xy>y AND y>0 Answer: IMPLIED BY

*x>1 ________ xy>y OR y>0 Answer: NONE

*x<1 ________ x$\le$1 Answer: NONE

*x<1 ________ x<0 Answer: IMPLIED BY

*x<1 ________ x+1 < 0 Answer: IMPLIED BY

*x<1 ________ x-1 < 0 Answer: IFF

*x<1 ________ x$^2$ < 1 Answer: NONE

*x<1 ________ x+y $\le$ y Answer: IMPLIED BY

*x<1 ________ xy < y Answer: NONE

*x<1 ________ x < z-x Answer: IFF

*x<1 ________ x < y OR y < 1 Answer: NONE

*x<1 ________ x < y AND y < 1 Answer: IMPLIED BY

*x<1 ________ xy < y OR y > 0 Answer: NONE

*x<1 ________ xy < y AND y > 0 Answer: IMPLIED BY

*x<1 ________ x < 0 OR x$^2$ < 1 Answer: IMPLIED BY

 A: 1: If $x>1$, then it is certainly true that $x\geq 1$. The converse is false ($1\geq1$, but it is not the case that $1>1$). So, the answer should be "IMPLIES".
7: You have ignored the possibility that $y$ is negative. Take $x=2$ and $y=-1$. Then $x>1$, but $xy=-2<-1$. So, the forward implication is false.  The backward implication is false as well: take $x=-1$ and $y=-2$. Then $xy=2>-2$, but $x\leq 1$. So, the answer should be "NONE".
13: If $x< 1$, then certainly $x\leq1$; the converse doesn't need to hold. So the answer should be "IMPLIES".
17: It is true that $x<1$ does not imply $x^2<1$; take $x=-2$, for instance.  However, the converse holds: $x^2<1$ is equivalent to $-1<x<1$, which implies $x<1$. So, this should be "IMPLIED BY".
20: If $x=0$ and $z=-2$, then $x<1$ but $0=x>-2=z-x$; so, the forward implication doesn't hold.  On the other hand, if $x=2$ and $z=5$, then $x<z-x$ but it is not true that $x<1$. So, this should be "NONE".
21: The forward implication is true: if $x<1$, there are two cases: either $y<1$ or $y\geq 1$. If $y<1$, then the RHS is true; if $y\geq 1$ and $x<1$, then certainly $x<y$, making the RHS true. The backward implication is false: if $y=3$ and $x=2$, then RHS is satisfied, but it is not true that $x<1$. So, this should be "IMPLIES".
25: If $x<1$, there are two cases: $x<0$ or $0\leq x<1$. In the first case, RHS is satisfied; in the second, $0\leq x^2<1$, and so RHS is still satisfied. So, the forward implication is true. For the backward: if $x<0$, then $x<1$; if $x^2<1$, then $-1<x<1$, and so $x<1$. So, 25 should be "IFF".
