what are the definitions of "ray" "segment", "line" as well as "intersection" and "union"?
Show that if A and B are distinct points, then the intersection of (ray(AB) and ray(BA)) = lineSegment(AB).
Show that if A and B are distinct points, then the union of (ray(AB) and ray(BA)) = the whole line (AB).
Can someone show these proofs. Thanks
I see only one reason for proving statements that are generally clear to preschool children, and that is to justify the statements in a very specific way, according to carefully chosen definitions and laws of reasoning. This is what a quote by Bertrand Russell from this thread talks about.
So, to bring people on the same page, you need to describe what type of microscope and other instruments you are using.The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere the in middle. Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using "simple" in a logical sense). And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics.
Mostly out of curiosity, I've tried to formalize the notion of "ray":
Similarly, we could arrive at segment AB by restricting and a line would result from T ranging across the reals.Given distinct points A, B, we denote ray(AB) by
Not that these will necessarily have anything to do with the OP's definitions... just wondering what you guys think about these proposed definitions!
In the Hilbert/Moore axiom system the undefined terms are: point, line, lie on, between,& congruent..
There are no numbers as such in synthetic geometry.
Now Ed Moise was one of Moore's PhD students. In his book he does introduce a Ruler Postulate by which he defines between in terms of coordinates. I found that this works best with undergraduates , participially mathed majors.