making two possibly overlapping sets become disjoint

Question

i'm self studying set theory, and would like help understanding the following statement regarding cardinal arithmetic for finite sets:

First consider addition. To add two cardinals k and l, the definition demands that we first select disjoint sets K and L with card K = k and card L = l. This is possible; if our first choices for K and L fail to be disjoint, we can switch to K x {0} and L x {1}.

The author states that K is equinumerous with K x {0}. What does K x {0} look like? The author only appears to use this syntax for ordered pairs, i.e. , which does not appear to be equinumerous with K. The author also only ever uses x⋅y for multiplication; however, is that what is implied here? For example {1,2,5}x{0}={0,0,0} vs {2,3,4}x{1}? So the sets are disjoint, but retain their cardinal value? What if L has 0 as a member?

This thread seems to answer my question, which is not presented in the book (at least so far). So instead of <{1,2,5}, {0}>, it becomes {<1,0>, <2,0>, <5,0>}? and so card K = card (K x {0}) = 3?

Thank-you!!

Answer 1

I believe $K \times {0}$ means the cartesian product.

Hence even though $\{ 1,2,5 \}$ and $\{ 2,3,4 \}$ are not disjoint.

we have $\{ 1,2,5 \} \times \{ 0 \}=\{ (1,0),(2,0),(5,0)\}$

and $\{ 2,3,4 \} \times \{ 1 \}=\{ (2,1),(3,1),(4,1)\}$

are disjoint due to different second coordinate.