Mathematical Proofs 102 aLet S be a collection of n2 numeric

Mathematical Proofs

10.2. (a)Let S be a collection of n>=2 numerically equivalent sets. Prove that these sets can be shown to be numerically equivalent by means of n-1 bijective functions between pairs of sets in S.

(b) what other questions is suggested by the problem in (a)?

Solution

Given there are n sets, let us number the set as A1,A2,...,An

Let us consider the sets are connected in the form of straight line chain as given below

A1----A2-----A3----A4....A(n-1)----An

There will be (n-1) bijective function namely from (A1-A2),(A2-A3) an so on

These will be bijective function

Reason: There must be each and every element covered in the A2, otherwise some elements will not be having mapping in the future sets
It must be one-one otherwise some elements will not be covered in the descendants sets

Hence the sets can be numerically equivalent by means of n-1 bijective functions between pairs of sets in S

It also proves that compositions of bijections are bijections each of the sets will have the same cardinality.