Cartesian Product and Bijection

[kidsasd987]

Homework Statement

Given two sets of Cartesian product

S=A₁×A₂...×A_n
P=(A₁×A₂...×A_n-1)×A_n

show that there exists bijection between the two sets.

Homework Equations

∀a₁,a₂:a₁∈A₁, a₂∈A₂: A₁×A₂=(a₁,a₂)

The Attempt at a Solution

let f be a function that maps

f: P → A₁×A₂...×A_n-1 where f((A₁×A₂...×A_n-1))∈A₁×A₂...×A_n-1 and f(A_n)∈A_nis this correct?

 

[member 587159]

No. You have to prove that there exists a bijection between the sets

##A_1 \times A_2 \times \dots A_{n-1} \times A_n## and ##(A_1 \times A_2 \times \dots A_{n-1}) \times A_n##, by giving an explicit bijection, or deducing the existence of such a function by other things you know.

These sets are not equal. The former contains elements of the form ##(a_1,\dots a_{n-1}, a_n)##, while the latter contains elements of the form ##((a_1,\dots, a_{n-1}),a_n)##. Formally, these are two different elements (but through the bijection you have to find, you can identify the two sets)

Always, when given such problems. Try the most obvious thing you can think of! This is:

Define ##f: A_1 \times A_2 \times \dots A_{n-1} \times A_n \to (A_1 \times A_2 \times \dots A_{n-1}) \times A_n: (a_1,\dots a_{n-1}, a_n) \mapsto ((a_1,\dots, a_{n-1}),a_n)##

Can you tell me why this is a bijection?