Question

Prove the theorem:

12.5 Theorem If α and β are transformations, then βα is a transformation.

Answer 1

ANSWER:

If α and β are transformations, then βα is a transformation. In other words we have to prove that, " The product of two transformations is itself a transformation ".

Proof :

Let α and β be two transformations.

Since for every point C there is a point B such that α(B) = C and for every point B there is a point A such that α(A) = B, then for every point C there is a point A  such that βα(A) = β(α(A)) = β(B) = C.

So βα is an onto mapping.

Also, βα is one-to-one, as the following argument shows.

Suppose βα(P) = βα(Q). Then β(α(P)) = β(α(Q)) by the definition of βα. So α(P) = α(Q) since β is one-to-one.

Then P = Q as α is one-to-one.

Therefore, βα is both one-to-one and onto.

Q.E.D.