The question provided appears to involve a set $A$ defined by both a base case and a recursive rule. Here's a structured breakdown of the problem:

Problem Breakdown:

1. Base Case:

The base of the recursion is given as: $28, 63 \in A$ This means that the numbers 28 and 63 are elements of the set $A$.

2. Recursive Rule:

The recursive rule is given as: $\text{If } a, b \in A, \text{ then } a + b \in A \text{ and } |a - b| \in A.$ This rule states that for any two elements $a$ and $b$ in the set $A$, both their sum $a + b$ and the absolute value of their difference $|a - b|$ must also be elements of $A$.

Analyzing the Options:

Given that the options provided in the image represent various expressions for the set $A$, let's analyze the set $A$ based on the given base case and recursive rule:

1. Base Elements:

   • Start with 28 and 63.

2. Recursive Application:

   • By the rule, if $a = 28$ and $b = 63$, then: $a + b = 28 + 63 = 91 \in A$ $|a - b| = |28 - 63| = 35 \in A$
   • Now, apply the rule again:
     • Using 28 and 35: $a + b = 28 + 35 = 63 \in A$ $|a - b| = |28 - 35| = 7 \in A$
     • Using 28 and 91: $a + b = 28 + 91 = 119 \in A$ $|a - b| = |28 - 91| = 63 \in A$
     • Using 63 and 91: $a + b = 63 + 91 = 154 \in A$ $|a - b| = |63 - 91| = 28 \in A$
   • Continue applying the rule to other pairs.

3. Closure under the Rule:

   • The repeated application of the rule $a + b$ and $|a - b|$ on any elements within $A$ suggests that $A$ will eventually include all multiples of 7.

Conclusion:

Based on the recursive construction, the set $A$ generated by the base elements 28 and 63 under the recursive rule will include all non-negative multiples of 7. Thus:

$A = \{7k : k \in \mathbb{N} \cup \{0\}\} = 7\mathbb{N}_0 = \{0, 7, 14, 21, 28, \dots\}$

This matches one of the simplifications shown in the image. Therefore, the correct interpretation of the set $A$ is the set of all non-negative multiples of 7.

Would you like to explore more about the problem or need further clarification on any steps?

Related Questions:

1. How do recursive definitions impact the structure of sets?
2. What role does the base case play in determining the final form of the set?
3. Can this recursive process be applied to generate other sets from different bases?
4. How do operations like $a + b$ and $|a - b|$ contribute to the closure of the set?
5. What mathematical concepts can help generalize this approach to other problems?

Tip:

Understanding recursive rules is crucial when working with sets, as they can reveal patterns and structures that are not immediately obvious. This approach is widely used in number theory and computer science.