To be as concrete as possible we can consider “free vectors” and “bound vectors” on the real line R (the usual number line which is in one-one correspondence with R and is, in fact, coordinatized by R.) [/FONT]In this case a bound vector is defined to be any ordered pair of real number (x1[/SIZE][/FONT], x2), e.g. (4, 5). The equivalence relation on bound vectors is (x1, x2) ~ (y1, y2) if and only if x1- x2 = y1 - y2.
- Verify that “~” is an equivalence relation.
Is it correct?
- x~x = reflexive -> 4 ~ 4
- x~y <=> y~x = symmetric -> 4 ~ 5 <=> 5 ~ 4
- x~y and y~z => x~z = transitive -> 4 ~ 5 and 5 ~ z => 4 ~ z