I plotted for all real number as a diagonal line on x-y axis and then plotted . By plotting , I have made sure none of the real number y is excluded. I found that all is possible on the diagonal line, but for x, I found it a very different story, namely, approaches neither of the axes--neither x nor y, which means that if , neither nor can be a zero.
If you take , you need not stick to because it did not say for all . Since means at least one, I do have one, namely x=1/2.
At any rate, it's very nice of you to help me. I enjoy reading many of you posts. You are quite good at math.
- this means that there is a single x, such that for every y that you pick, xy=1. So, logically, how do you disprove such a statement? you simply find one such y, for which .
This is the essence of the proof - you assume by contradiction that there exists such an x. That means that for every y that you pick, xy=1 should be true.
Specifically, it should be true for y=1. That means 1*x = 1 -> x = 1 (remember, one x for all y).
So you have that x*y = 1*y = 1 for all real y. To prove that is not true, simply take any other y - for example y2 = 2. Your assumption implies that x*y2 = 1*y2 = 1, but 1*y2 = 1*2 = 2. So you get 2=1, which is obviously a contradiction.