CARDINALITY AND SET THEORY QUESTION Prove that there is a bi

CARDINALITY AND SET THEORY QUESTION: Prove that there is a bijection between A x B x C and (A x B) x C

Solution

There is a natural bijection from AxBxC to (AxB)xC

f((a,b,c))=((a,b),c)

Let,

Let, ((x,y),z) be in (AxB)xC

So, f((x,y,z))=((x,y),z)

Hence, f is surjective

Let, f((a,b,c))=f((e,f,g))

((a,b),c)=((e,f),g)

Hence,

a=e

b=f

c=g

Hence, f is injective and hence a bijection.