Define a set X recursively as follows.
B. 2 ∈ X.

R. If x ∈ X, so is x + 10.

Use induction to prove that every element of X is even.

To prove that every element of \( X \) is even, let's proceed with mathematical induction based on the recursive definition of \( X \).

### Definition Recap
1. **Base case (B):** \( 2 \in X \).
2. **Recursive rule (R):** If \( x \in X \), then \( x + 10 \in X \).

We need to prove that every element \( x \in X \) is an even number.

### Proof by Induction
#### Step 1: Base Case
The base element \( 2 \in X \) is clearly even.

#### Step 2: Inductive Hypothesis
Assume that for some \( k \in X \), \( k \) is even. In other words, \( k = 2n \) for some integer \( n \).

#### Step 3: Inductive Step
By the recursive rule (R), since \( k \in X \), we have \( k + 10 \in X \).
Now, let's show that \( k + 10 \) is also even:
\[
k + 10 = 2n + 10 = 2(n + 5),
\]
which is clearly an even number (as it is divisible by 2).

#### Conclusion
By induction, every element of \( X \) is even.

This completes the proof.

Do you have any questions, or would you like more details? 

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Here are some related questions to explore:
1. How would the proof change if the recursive rule was \( x + 5 \) instead of \( x + 10 \)?
2. What would happen if the initial element in \( X \) was an odd number?
3. Can we use induction to show all elements of \( X \) are divisible by any constant multiple?
4. How would we modify the proof if the base case was \( 4 \in X \) instead of \( 2 \)?
5. What if there were two base cases, like \( 2 \in X \) and \( 4 \in X \)?

**Tip:** In recursive proofs, always verify if each added element maintains the required property to simplify induction.

Define a set X recursively as follows. B. 2 ∈ X. R. If x ∈ X, so is x + 10. Use induction to prove that every element of X is even.

Explore a detailed proof using mathematical induction to show that every element in a recursively defined set X is even. Given that 2 ∈ X and if x ∈ X, then x + 10 is also in X, this proof illustrates core concepts of induction, recursion, and even numbers. Suitable for Grades 10-12.

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