
Simple Graphs and Paths
This might seem easy, but im having trouble putting my proofs into words.
Let G = (V,E) be simple graph,
I need to prove for all vertices (x,y,z) of G, the follwoing are true:
1) There is a path from x to x.
2) If there is a path from x to y then there is path y x.
3)If there is a path from x y, and yz then there is path xz.
Now im not sure whats expected when proofing this.
Do i just say a path from x to x exists since its a path to itself?
and...
if a path xy exists then there is a path y x because x and y are two connected vertices by a edge?
and..
if there is a path from xy, and yz then there is a path xz since vertices x and y are connected by an edge, and yz is connected by an edge, this provides a path from vertices x to z through vertices y.
Id like some feedback if possible thanks.

Just work from the definitions and find a path that satisfies the conditions.
1: You can take the vertice sequence {x} which starts and stops at x and exists by axiom.
2: if F: (a,b,...,z) is the path from x to y, then G: (z,...,b,a) is the path from y to x, which exists because every edge q in G exists by assumption since it exists in F.
3: if F: (a,b,...,m), G: (n,o...,z) are the respective paths from x to y and from y to z, then the path H:(a,b,...,m,n,o,...,z) is the path from x to z, and all edges exist because they existed in F and G.
I couldn't be arsed with indices of the vertices, which you should use if you do this for homework, but using the alphabet is clearer.
General rule of thumb when solving these things: don't think too deeply and just construct the answers straight from the definition and question.