Prove: For every pair of points $P, Q$
1. $d(P,Q)>0$
2. $d(P,Q) = 0$ if and only if $P=Q$
3. $d(P,Q) = d(Q,P)$
where $d(P,Q)$ is defined as the distance between $P$ and $Q$ and $d$ is a function
For the first part, I tried to assume the contrary, but I'm not sure where this gets me.
On
The three properties to prove that $d$ satisfies comprise the definition of a distance or metric. If $d$ is a distance, then it automatically satisfies these properties by definition. I think you're referring to a specific proposed $d$, which from the tag of "euclidean-geometry" I assume to be Euclidean distance. For the third property, exchange $P$ and $Q$ and mechanically show you obtain the same expression for $d(P,Q)$ as you get for $d(Q,P)$. For the second property,start with the assumption that $P = Q$ and evaluate $d(P,Q)$ directly. You'll notice this is exceptionally straightforward. Next, start with the assumption that $d(P,Q) = 0$. Then, directly derive a relationship between $P$ and $Q$. Finally, I think the first property should say either $d(P,Q) \geq 0$ (which is the standard definition) or it should say $P \neq Q \rightarrow d(P,Q) > 0$ (which also makes for an equivalent definition along with the second property), but it should not say $d(P,Q) > 0$.
In analysis or a topology or an abstract math course the idea of distance function is an abstract concept:
Let S be any set then a "metric" or a "distance" is a function, $d:S \times S \rightarrow \mathbb R$, a function that maps a pair of elements of the set S to a real number, such that.
1) $d(x,y) \ge 0$
1a) $d(x,y) = 0 \iff x = y$
2) $d(x, y) = d(y,x)$ for all $x,y \in S$
3) $d(x,z) \le d(x,y) + d(y,z)$.
Now this is an abstract definition. S could be a set of singing frogs and d could measure when their songs first start to differ, for all we know. (Assuming all songs start on the same note and all songs are eventually different.)
Now the fact that we call this function a "distance" should be a hint to what we have in mind.
So the question is (I'm assuming): prove that in Euclidean plane geometry the actual common sense use of distance between points fits the abstract definition of a metric.
The only problem is, that in our high school geometry class, we can't "prove" these statements about points; we were given them as axioms.
So:
1) Prove $d(P,Q) \ge 0$. Well this is one of Euclids basic postulates (the 1st, I believe) that between any two points there is a "length". In other words there is a distance between any two points; no two points can't be measured, distances are positive real numbers, distances between two points are consistant-- one day it won't suddenly be that New York is suddenly two and half miles from San Francisco; etc. But we can't prove these. This is a basic axiom.
2) Prove $d(P,Q) = 0 \iff P = Q$: Well, any two different points are always some positive distance apart. A point itself is no distance from itself. (Again, an axiom. As in ... well, it doesn't make sense otherwise..)
3) Prove $d(P,Q) = d(Q,P)$: Distance is measured between endpoints. It doesn't matter which end point we measure from. The is Euclid's definition of distance. It is defined as being between a pair; not an ordered pair-- an unordered pair.
At least I'm assuming that is what the question means.