# Homework Help: Bijection between products of countable sets

1. Oct 15, 2011

### The1TL

1. The problem statement, all variables and given/known data
Let S1 = {a} be a set consisting of just one element and let
S2 = {b, c} be a set consisting of two elements.

Show that S1 × Z is bijective to S2 × Z.

2. Relevant equations

3. The attempt at a solution

So I usually prove bijectivity by showing that two sets are equinumerous, But in this case S1 and S2 are not so that makes it more difficult.

2. Oct 15, 2011

### spamiam

You're right, this problem demonstrates how things can become unintuitive when dealing with infinite sets. If A were any nonempty finite set, this claim would be false, since in that case
$$|S_1 \times A| = |S_1||A| = |A|$$
$$|S_2 \times A| = |S_2||A| = 2|A| \; .$$

However, since $\mathbb{Z}$ is infinite, we have more leeway. Can you think of a proper subset of $\mathbb{Z}$ that is equinumerous to $\mathbb{Z}$?

3. Oct 16, 2011

### Deveno

here is one idea:

.....
.....
(a,-3) <--> (c,-2)
(a,-2) <--> (b,-1)
(a,-1) <--> (c,-1)
(a,0) <--> (b,0)
(a,1) <--> (c,0)
(a,2) <--> (b,1)
(a,3) <--> (c,1)
(a,4) <--> (b,2)
.....
.....

can you prove this is, in fact, a bijection?