# For any n, m given ##I_n \thicksim I_m##, prove that ## n=m##

• issacnewton
In summary, this conversation discusses the proof that for all natural numbers n and m, if ##I_n \thicksim I_m##, then ## n=m##. The proof involves showing that if ##n < m## or ##n > m##, there is a contradiction, and therefore, it must be that ##n = m##.
issacnewton
Homework Statement
Prove that for all natural numbers n and m, if ##I_n \thicksim I_m##, then ## n=m##.
Relevant Equations
Definition of equinumerous sets and definition of bijection
Let ##n, m## be arbitrary in ##\mathbb{N}##. Assume that ##I_n \thicksim I_m##. We are given that

$$I_n = \big\{ i \in \mathbb{Z}^+ | i \leq n \big\}$$
$$I_m = \big\{ i \in \mathbb{Z}^+ | i \leq m \big\}$$

These are following sets

$$I_n = \bigg \{ 1,2,3,\ldots, n \bigg \}$$
$$I_m = \bigg \{ 1,2,3,\ldots, m \bigg \}$$

Now, assume that ##n < m##. Since ##I_n \thicksim I_m##, there is a bijection from ##I_n## to ##I_m##. Since there are less members in ##I_n## than in ##I_m##, we can map these ##n## members to first ##n## members in ##I_m##. Then we are left with ##m-n## members in ##I_m## which do not have any pre image in ##I_n##. But since the function is onto, this is a contradiction and hence ## n \nless m ##. Similar argument can be given when ##n > m##. So we also prove that ## n \ngtr m##. So, we must have ##n = m##. Is this a valid proof ?

IssacNewton said:
Homework Statement:: Prove that for all natural numbers n and m, if ##I_n \thicksim I_m##, then ## n=m##.
Relevant Equations:: Definition of equinumerous sets and definition of bijection

Let ##n, m## be arbitrary in ##\mathbb{N}##. Assume that ##I_n \thicksim I_m##. We are given that

$$I_n = \big\{ i \in \mathbb{Z}^+ | i \leq n \big\}$$
$$I_m = \big\{ i \in \mathbb{Z}^+ | i \leq m \big\}$$

These are following sets

$$I_n = \bigg \{ 1,2,3,\ldots, n \bigg \}$$
$$I_m = \bigg \{ 1,2,3,\ldots, m \bigg \}$$

Now, assume that ##n < m##. Since ##I_n \thicksim I_m##, there is a bijection from ##I_n## to ##I_m##. Since there are less members in ##I_n## than in ##I_m##, we can map these ##n## members to first ##n## members in ##I_m##. Then we are left with ##m-n## members in ##I_m## which do not have any pre image in ##I_n##. But since the function is onto, this is a contradiction and hence ## n \nless m ##. Similar argument can be given when ##n > m##. So we also prove that ## n \ngtr m##. So, we must have ##n = m##. Is this a valid proof ?
Looks fine to me.

Thanks Mark

## 1. What does the statement "I_n is equivalent to I_m" mean?

The statement "I_n is equivalent to I_m" means that the two sets I_n and I_m have the same number of elements. In other words, they have a one-to-one correspondence, meaning that each element in I_n can be paired with a unique element in I_m and vice versa.

## 2. How can I prove that n=m if I_n is equivalent to I_m?

To prove that n=m, you can use a proof by contradiction. Assume that n≠m and show that this leads to a contradiction. This can be done by showing that if n≠m, then I_n and I_m cannot have a one-to-one correspondence, which contradicts the given statement that I_n is equivalent to I_m.

## 3. Can you provide an example to demonstrate this statement?

Yes, for example, let n=3 and m=5. The set I_n would be {1,2,3} and the set I_m would be {1,2,3,4,5}. It is evident that these two sets do not have the same number of elements, and therefore, they are not equivalent. However, if n=m=3, then both sets would be {1,2,3}, and they would be equivalent.

## 4. Is it possible for two sets to have the same number of elements but not be equivalent?

No, if two sets have the same number of elements, then they are equivalent. This is because having a one-to-one correspondence is a necessary and sufficient condition for two sets to be equivalent.

## 5. How is this statement useful in mathematics?

This statement is useful in many areas of mathematics, such as set theory, combinatorics, and number theory. It allows us to compare the sizes of different sets and prove various mathematical theorems. Additionally, it is a fundamental concept in understanding cardinality and infinity.

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