I am reading Armstrong: Groups and Symmetry Chapter 12: Partitions
In talking about partitions and equivalence classes Armstrong writes on Page 60 (see attachment for the relevant pages of Armstrong's book):
Let X be a set and R be a subset of the cartesian product X x X. In other words, R is a collection of ordered pairs (x,y) whose coordinates x,y come from X. Given two points x and y of X we shall say that x is related to y if the ordered pair (x,y) happens to lie in R. If properties (a), (b) and (c) [properties of equivalence class - see Armstrong page 60 attached] are valid, then we call R an equivalence relation on X. For each x the collection of points which are related to it irs written and called the equivalence class of x.
(12.1) Theorem R(x) = R(y) whenever (x,y) R
Although Armstrong writes "If properties (a), (b) and (c) are valid, then we call R an equivalence relation on X" he clearly believes that R is an equivalence relation and the Theorem seems to be a statement of this.
My questions are as follows:
If R actually is an equivalence relation then we have
x is related to y iff the ordered pair (x,y) happens to lie in R
Now if x is related to y then if R is an equivalence relation then y must be related to x - that is the ordered pair (y,x) must lie in R
BUT (x,y) can lie in R and (y,x) might not - so R does not seem to be an equivalence relation?
Second question: Is Theorem 12.1 above a statement that R is an equivalence relation. If so, how is the statement of the theorem equivalent to R possessing the three defining properties of an equivalence relation.
Can someone please help?