To determine whether the sets $\{3, 4, 5\}$ and $\{9, 31, 20\}$ are equivalent (denoted by "$\sim$"), we need to understand the criteria for set equivalence.

Two sets are considered equivalent if and only if there is a one-to-one correspondence (bijection) between their elements, meaning that both sets must have the same number of elements, and their elements can be matched uniquely with one another, although the specific values do not need to be the same.

Step-by-step analysis:

1. The set $\{3, 4, 5\}$ has 3 elements.
2. The set $\{9, 31, 20\}$ also has 3 elements.

Since both sets have the same number of elements, it is possible to establish a one-to-one correspondence between their elements. Therefore, the sets are equivalent.

Conclusion:

The statement $\{3, 4, 5\} \sim \{9, 31, 20\}$ is true because both sets have the same number of elements and are thus equivalent.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the difference between set equivalence and set equality?
2. Can two sets with different numbers of elements be equivalent?
3. How do you define a one-to-one correspondence between sets?
4. Is set equivalence affected by the specific values of the elements?
5. What is the difference between a set and a multiset?

Tip: Remember, set equivalence doesn't require the elements to be the same, only that there is a matching number of elements!