Let's classify each statement:

1. "Let $x$ then $y$."

• Select: is a Definition
  This is a definition because it sets a specific relationship between $x$ and $y$.

2. "For $a$, we set $b$."

• Select: is a Definition
  This defines $b$ in terms of $a$, which is characteristic of a definition.

3. "For $c$, we set $d$."

• Select: is a Definition
  Similar to the previous statement, this establishes a specific relationship or assignment.

4. "There are infinitely many prime numbers."

• Select: is a Theorem
  This is a theorem because it has been proven to be true (Euclid's theorem).

Summary:

1. Definition
2. Definition
3. Definition
4. Theorem

Do you want details or have any questions? Here are some related questions to consider:

1. What distinguishes a definition from a theorem?
2. Can a conjecture become a theorem?
3. What is an example of a well-known theorem in mathematics?
4. How do definitions help in constructing mathematical proofs?
5. What are some famous conjectures in mathematics that have yet to be proven?

Tip: When studying mathematical statements, focus on their context to determine whether they define terms, assert truths (theorems), or propose ideas (conjectures).