Let $f :\mathbb R \to \mathbb R$ be a function. Then
Show that $f$ is one one if the graph of $f$ intersects any line parallel to the $X$ - axis in at most one point.
Show that $f$ is onto if the graph of $f$ intersects every line parallel to the $X$ - axis.
My attempt : If I take $f(x) = y = x$ then above statements trivially holds. But this is not a good way and we can't prove any theorem by taking an example.
I don't know how to prove it.
Please help me.
Let D, with equation y = a be a line parallel to the X axis.
Suppose that the graph of function f intersects D at more that 1 point, say 2 distinct points.
The Y-coordinate of these points is a, since these points are on D.
Their X cordinates are 2 different x values, since these 2 points are distinct, say x1 and x2.
The cordinates are therefore ( x1, a) and (x2, a).
It means that f(x1) = a and f(x2) = a, with x1 different from x2.
But that case never occurs with a one-to-one function, that is, a function such that no two distinct elements of the domain have the same image.
Out of this, one can conclude that the contrapositive of the sentence you have to prove is true.
But, if the contrapositive of a sentence is true, then this sentence is also true.