1
$\begingroup$

Prove that if $\overline{AB} = \overline{CD}$, then either $A = C$ and $B=D$ or $A = D$ and $B = C$.

I use the definition of a segment: \begin{align*} &\overline{AB} = \{A,B\} \cup \{ P \; | \; A * P * B\} \\ &\overline{CD} = \{C,D\} \cup \{ P \; | \; C * P * D\} \end{align*} Thus, if $\overline{AB} = \overline{CD}$ we must have \begin{equation} \{A,B\} \cup \{ P \; | \; A * P * B\} = \{C,D\} \cup \{ P \; | \; C * P * D\} \end{equation} Can I conclude directly from the equality above that either $A=C$ and $B=D$ or $A=D$ and $B=C$? If not, where do I go next to complete the proof?

  • 0
    Since you doubt this is sufficient, chances are that it is insufficient. A mathematical argument should be such that you are 100% certain of the result after reading it. – 2017-01-24
  • 0
    Perhaps you meant $\overline{AB} \cong \overline{CD}$ (instead of just $\overline{AB} = \overline{CD}$)? – 2017-01-24
  • 0
    @JoseArnaldoBebitaDris: I guess it's equality in the sense of same set of points. – 2017-01-24
  • 0
    @MvG: Yes, but it can also be interpreted to mean equality of distances between the endpoints of the (two) line segments, which is not the same thing. – 2017-01-24
  • 1
    Perhaps the best way to deal with this question is to (1) define the *interior* of a line segment, (2) define the *endpoints* of a line segment, and (3) show that the endpoints of two congruent line segments coincide (which consists of two cases). – 2017-01-24
  • 1
    If $A$ and $B$ are both in $S:=\{P|C*P*D\}$ then $\overline{AB}$ is a proper part of $\overline{CD}$ so this is impossible. WLOG suppose $A$ is not in $S$. Then either $A=C$ or $A=D$. WLOG suppose $A=C$. Now we can't have $B\in S$ because, again, $\overline{AB}$ would be a proper part of $\overline{AD}=\overline{CD}$. Hence $B\not\in S$ and so $B=D$ (supposing $A\neq B$). – 2017-01-24

1 Answers 1

1

In general the question of what one can conclude directly is a matter of context. If you have a sufficiently experienced audience, you may assume that they will be able to fill in the gaps in a high-level proof, while for a less experienced context you'll have to fill in more of these gaps yourself. And for formal proof verification you usually have to use very small steps. But the message here is that you as the one presenting the proof should be able to fill in the gaps. If you have a child sitting in the audience which keeps asking “why”, you should be able to continue till you conclude each branch of the agument by citing some axiom or established deduction rule.

If even you yourself can't say why that conclusion should hold (which I assume is the case else you'd not be asking), then no, it is not a good idea to make that conclusion in a single step and hope that either your audience can fill in the gaps you can't, or will trust you even though you are unsure yourself.

So what do you need here? I'd say you should observe that the endpoints can not be seen as inner points, so they are distinguised. Go for a proof by contradiction. Assume $X*A*Y$ for some $\{X,Y\}\subset\overline{AB}$. In words, assume that the endpoint is part of the segment defined by two other points on the segment (which might include $A$ and/or $B$). Once you have disproven this, you know that the segment, taken as a set of points, has two distinguished endpoints, since no other pair of points can define all points along the segment, and in particular will fail to include $A$ and $B$.