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Problem Summary:

We are given the set $A = \{17, 19, 31, 37, 43\}$. The task is to: (a) Find the total number of proper subsets of $A$.
(b) Find the total number of subsets of $A$.

Solution:

(b) Total number of subsets of $A$:

The formula to find the total number of subsets of a set with $n$ elements is:

$\text{Total subsets} = 2^n$

Here, $n = 5$ (since $A$ has 5 elements). Therefore:

$\text{Total subsets} = 2^5 = 32$

(a) Total number of proper subsets of $A$:

A proper subset is any subset of $A$ that is not equal to $A$ itself. The number of proper subsets is:

$\text{Proper subsets} = \text{Total subsets} - 1$

This is because we exclude the set $A$ itself from the total subsets. So:

$\text{Proper subsets} = 32 - 1 = 31$

Final Answers:

(a) Total number of proper subsets of $A$: 31
(b) Total number of subsets of $A$: 32

Would you like further explanation or details on subsets and proper subsets?

Here are five related questions for practice:

1. What is the total number of subsets of a set with $n = 6$ elements?
2. How many proper subsets does a set with $n = 7$ elements have?
3. What is the difference between a proper subset and a subset?
4. Can an empty set be a proper subset of any set? Why?
5. If a set has $n = 10$ elements, how many subsets and proper subsets does it have?

Tip: To find subsets quickly, remember that the total subsets grow exponentially with the size of the set: $2^n$.