Let's break down the question:

You are asked to find:

• a) How many subsets does the set $A = \{5, 6, 7, 8, 9, 10\}$ have?
• b) How many proper subsets does $A$ have?

a) Number of subsets:

The number of subsets of a set is given by $2^n$, where $n$ is the number of elements in the set.

In this case, $A$ has 6 elements: $5, 6, 7, 8, 9, 10$.

So, the number of subsets is: $2^6 = 64$ Thus, set $A$ has 64 subsets.

b) Number of proper subsets:

A proper subset of a set is a subset that is not equal to the set itself. To find the number of proper subsets, we subtract 1 from the total number of subsets (because the set itself is not a proper subset).

Therefore, the number of proper subsets of $A$ is: $64 - 1 = 63$

Final answers:

• a) $A$ has 64 subsets.
• b) $A$ has 63 proper subsets.

Would you like more details on how this works? Here are some related questions you might find useful:

1. What is the difference between a subset and a proper subset?
2. How does the formula $2^n$ for subsets apply to different sets?
3. Can a set have more than one proper subset?
4. What is the concept of the power set and how does it relate to subsets?
5. How do the number of subsets and proper subsets change if the set size increases?

Tip: When dealing with subsets, remember that the power set of a set includes all possible subsets, including the set itself. Proper subsets exclude the set itself.