a Give the definition of an equivalence relation on a set A

(a) Give the definition of an equivalence relation ~ on a set A. (b) Let A be the set R^n without the zero vector. Define a relation ~ on A in the following way: v ~ w whenever there exists a non-zero c element of R such that w = cv, for v, w element of A. Show that ~ is an equivalence relation on A.

Solution

(a) Let ~ be a relation on a set A. ~ is said to be an equivalence relation if it satisfies the following

three properties:

(i) (Reflexivitiy): For all x in A, x ~x. (Every element is related to itself)

(ii) (Symmetry) : x~y implies y~x

(iii) (Transitivity) : x~y and y~z implies x ~z.

(b) Now let A = Rⁿ -{0}.

By definition, v~w if there exists a (necessarily non-zero) scalar c such that w =cv.

Claim: ~ is an equivalence relation:

(i) Reflexivity: Take c =1 ; v ~v as v =1.v

(ii)Symmetry: w =cv implies v = c^-1 w. (c is non-zero)

(iii) Transitivity: Let u ~v , and v~w. Then there exist non-zero b and c such that

                              v =bu and w =cv (by definition of ~)

this imples w =cv=cbu . Now cb is non-zero as c and b are both non-zero.

So w~u and hence ~ is transitive.

Thus ~ is an equivalence relation on A.