Question

If two sets are infinite then they are equivalent, true or false

Answer 1

false

Answer 2

False

Answer 3

Why is that?

Answer 4

e.g.:
The set of all odd numbers
and the set of all even numbers

Answer 5

Can you give a different reason?

Answer 6

The set of rational numbers
and the set of irrational numbers

Answer 7

whatever polpak says :)

Answer 8

So it's because they dont have the same number of elements?

Answer 9

Two sets cannot be 'equivalent' if they don't contain the same elements. There are many infinite sets that contain different elements.

Answer 10

Now if two sets are equivalent, are they infinite?

Answer 11

I'm not sure what you think equivalent means..

Answer 12

Generally you'd say they are equivalent with regard to a particular operation.

Answer 13

http://mathforum.org/library/drmath/view/65122.html

Answer 14

My question just says if two sets are equivalent, are they infinite? true or false.

Answer 15

According to that definition:

Sets are called equivalent when they have the same cardinality (number
of elements). What's happening here is that when we ask whether two
sets are equivalent, we are ignoring the names of the elements, and
considering only what makes one set essentially different from
another, which is the number of elements it has. So any set of six
elements is considered equivalent to the one you were given.

Answer 16

Under that definition, the set of even numbers and the set of odd numbers have the same cardinality.

Answer 17

Oh. That seems silly, but ok. Then if the sets have the same cardinality they would be considered equivalent

Answer 18

So if two sets are equivalent, then they are infinite?

Answer 19

I would say that they're equivalent with respect to their cardinality, but perhaps I'm just picky.

Answer 20

No

Answer 21

It goes on and says this:

In other words, your question is really about the definition of
'equal' and 'equivalent' in this context. Equivalent sets contain the
same number of elements, while equal sets contain not only the same
number of elements, but also the exact same elements, regardless of order.

Answer 22

So if two sets are infinite they are equivalent?

Answer 23

well no - there are different levels of inifinity:

http://en.wikipedia.org/wiki/Cardinality

Answer 24

as long as they are both the same kind of infinite yes. But if one is uncountably infinity and one is countable, then they are not equivalent.

Answer 25

what polpak said :)

Answer 26

for example:

cardinality of the Real numbers is greater than that of the natural numbers

Answer 27

One of Cantor's most important results was that the cardinality of the continuum (\mathfrak{c}) is greater than that of the natural numbers (ℵ0); that is, there are more real numbers R than whole numbers N.

Answer 28

So FALSE: if two sets are equivalent, then they are infinite. Because there are different levels of infinity?

Answer 29

Two sets can be equivalent (or not), whether they are finite or infinite.


The original question was:
If two sets are infinite then they are equivalent, true or false

And the answer is FALSE:

For example:
The set of counting numbers have a lower cardinality (measure of infinity) than the real numbers.

Answer 30

Okay I understand that now. My second question was if two sets are equivalent, then they are infinite. True or False. Wouldnt that be false?

Answer 31

FALSE:

These two sets are equivalent and finite:

{1,2,3}
and
{4,5,6{

Answer 32

that last bracket got messed up :)

Answer 33

So its false because there are different kinds of infinity right?

Answer 34

If two sets are equivalent, then they are infinite --> FALSE

Because the two sets below are equivalent but they are not infinite:

set1={1,2,3}
set2={4,5,6}

Answer 35

so in the above example we are not discussing levels of infinity (cardinality) at all.
we have a counter example, of two finite sets that are equivalent (same size) and they are not infinite ---> FALSE.

Answer 36

I have to state a reason

Answer 37

The reason is:
There are pairs of finite sets which are equivalent.

Answer 38

Read this question, and it's answer:

http://mathforum.org/library/drmath/view/65122.html

Answer 39

That didnt really help me :/

Answer 40

sets are considered equivalent if they have the same number of elements (also called the set cardinality). right ?

Answer 41

Yes, I understand that

Answer 42

Is the cardinality of this set {1,2,3} the same as the cardinality of the set {4,5,6} ?

Answer 43

yes

Answer 44

So these are examples of finite sets that have the same cardinality.

Answer 45

Okay

Answer 46

which means they are equivalent

Answer 47

And this contradicts the statement that "if two sets are equivalent, then they are infinite" - coz we just provided examples that disprove this.

Answer 48

Yes