Decide if each of the following statements is a definition, a theorem or a conjecture.

selectis a Conjectureis a Definitionis a Theorem

**1.** Let  then .

selectis a Conjectureis a Definitionis a Theorem

**2.** For  we set .

selectis a Conjectureis a Definitionis a Theorem

**3.** For  we set .

selectis a Conjectureis a Definitionis a Theorem

**4.** There are infinitely many prime numbers.

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).

Decide if each of the following statements is a definition, a theorem or a conjecture: 1. Let x then y. 2. For a, we set b. 3. For c, we set d. 4. There are infinitely many prime numbers.

Explore the classification of mathematical statements as definitions, theorems, or conjectures. This resource covers key concepts including the relationship between variables and Euclid's theorem on the infinitude of prime numbers. Suitable for students in Grades 9-12.

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