Question

Give a counterexample for the following statement:
\( \overline{\overline A } \leq \overline{\overline B } \text{ implies } A \subset B \)

Answer 1

What does \(\overline{\overline{S}}\) mean? Is that the cardinality of the set?

Answer 2

yes

Answer 3

\(A=\mathbb{Z}\)
\(B=\mathbb{R}\times \mathbb{R}\)

Answer 4

There are numerous counterexamples.

Answer 5

Well \( \mathbb{Z}\) is a subset of \( \mathbb{R} \)

Answer 6

The cartesian product of two sets is a set of ordered pairs.

Answer 7

Can you explain to me what cardinality actually means

Answer 8

how about B is all even numbers
and A is the first ten odd number?
(I don't know if you need infinite sets or what)

Answer 9

not sure either. i am just trying to figure out all this stuff

Answer 10

but I am assuming that it refers to denumerable sets

Answer 11

No, uncountable sets still have cardinalities.

Answer 12

I have never taken this kind of math, but cardinality as I basically understand it is how many elements it has (I don't know about sets of ordered pairs)

Answer 13

Ordered pairs act as elements of the set. They do not change anything.

Answer 14

so what's the matter with my finite example?

Answer 15

Nothing!

Answer 16

oh, sweet :)

Answer 17

ok so how abt this example:
A= \( \mathbb{Q^-} \)
B= \( \mathbb{Q^+} \)

Answer 18

hahahahahah

Answer 19

like do the cardinalities equal each other?

Answer 20

Yes.

Answer 21

they are both\[\aleph_0\]right? (side question)

Answer 22

lol I think soooo

Answer 23

Yep. They can both be put on a one-to-one correspondence with the positive integers.

Answer 24

I gotta take this formally some day...
thanks!

Answer 25

THANKS GUYSSSS :))