Definition of a relation generated by R - Can someone explain this definition and give a simple example?

Question

From a footnote in Rotman's Algebraic topology:

https://s3.postimg.org/i2739kx8z/QQQ.png

Could someone explain what exactly this definition is saying and give a simple example?

It seems like it's saying given a finite list of $n$ elements, $n-1$ of them are; equal, or in the previous relation in some order. But this doesn't seem to make sense.

As an example:

$X = \{1,2,3, 4\}$ with less than relation $R = \{(1,2), (1,3), (2,3), (2,4), (3,4)\}$,

what would be an example of $R'$?

Answer 1

It is stated that $R^\prime$ derived from $R$ via the given rules is an equivalence relation. So we can check if reflexivity, symmetry and transitivity is provided by these rules.

• Reflexivity:

  With $(x,y)\in R$ we obtain $(x,x)\in R'$ and $(y,y)\in R'$ since we can apply $x_i=x_{i+1}$ with $i=0$. This way it is assured that all elements of the diagonal are elements of $R'$. $$\{(x,x)|x\in X\}\subseteq R'$$

• Symmetry:

  Whenever $(x,y)\in R'$ we also need $(y,x)\in R'$. This is assured by the rules $(x_i,x_{i+1})\in R$ or $(x_{i+1},x_i)\in R$.

• Transitivity:

  Is given by the two rules to obtain symmetry and reflexivity above together with the chaining of the elements with index range $0\leq i\leq n-1$.