suppose b is a proper subset of c. a. if n(c) = 8, what is the maximum number of elements in b? b. if n(b) = 8, what is the maximum number of elements in c? c. what is the maximum number of elements that b could have? a. the maximum number of elements in b is blank. b. there is no limit on how many elements b could have.

Question

Accepted Answer

### Part a: If $ n(C) = 8 $, what is the maximum number of elements in $ B $?
# Explanation:
## Step 1: Recall proper subset definition
A proper subset $ B $ of $ C $ means $ B \subseteq C $ and $ B
eq C $. So $ B $ can contain at most all elements of $ C $ except one.
## Step 2: Calculate maximum size of $ B $
If $ n(C)=8 $, then the maximum number of elements in $ B $ (as a proper subset) is $ 8 - 1=7 $.
# Answer:
$ 7 $

### Part b: If $ n(B) = 8 $, what is the maximum number of elements in $ C $?
# Explanation:
## Step 1: Recall subset relationship
Since $ B $ is a proper subset of $ C $, $ C $ must contain all elements of $ B $ and at least one more element.
## Step 2: Determine maximum size of $ C $
There's no upper limit mentioned, but if we consider the context (assuming finite sets, but the problem doesn't restrict), however, if we think in terms of $ B $ being a proper subset, $ C $ can have infinitely many elements, but if we assume finite and the question is about maximum (which is unbounded in general), but if we misread (maybe typo, but as per proper subset: if $ B $ has 8 elements, $ C $ must have at least 9, but there's no maximum finite number. But maybe the question is reversed? Wait, no. Wait, the original problem: "If $ n(B) = 8 $, what is the maximum number of elements in $ C $?" But since $ B $ is a proper subset of $ C $, $ C $ must have more elements than $ B $, but there's no upper bound (unless restricted). But maybe the question was "minimum"? If it's maximum, there's no limit. But looking at the options (A: maximum number of elements in B is [blank], B: no limit). Wait, for part b, if $ n(B)=8 $, and $ B $ is a proper subset of $ C $, $ C $ can have any number of elements greater than 8, so there's no maximum (infinite or unbounded). So the answer for part b (the second question in the image: "What is the maximum number of elements that B could have? Wait, no, the third question: "What is the maximum number of elements that B could have?" Wait, maybe the image has a typo. Wait, re - examining the image:

The first two questions:

a. If $ n(C)=8 $, max elements in $ B $ (proper subset) → 7.

b. If $ n(B)=8 $, max elements in $ C $ → but $ B $ is proper subset of $ C $, so $ C $ must have more than 8, but no maximum (so option B: "There is no limit on how many elements B could have" – wait, maybe the question was mislabeled. If the third question is "What is the maximum number of elements that B could have?" and if there's no restriction on $ C $, but if $ B $ is a proper subset of some set, but if $ C $ can be any set, then $ B $ can have any number of elements (no limit). But for the first part (a), when $ n(C)=8 $, the maximum is 7.

Let's focus on the first sub - question (a):

# Explanation for part a:
## Step 1: Proper Subset Definition
A proper subset $ B $ of a set $ C $ satisfies $ B\subseteq C $ and $ B
eq C $. This means $ B $ can contain all elements of $ C $ except at least one element.
## Step 2: Calculate the Maximum Size of $ B $
Given that the number of elements in $ C $, $ n(C) = 8 $. To find the maximum number of elements in $ B $ (a proper subset), we subtract 1 from $ n(C) $ because $ B $ cannot be equal to $ C $. So $ n(B)_{\text{max}}=n(C)- 1$.
Substituting $ n(C) = 8 $, we get $ n(B)_{\text{max}}=8 - 1=7 $.

# Answer for part a:
$ 7 $

For the second sub - question (b):

# Explanation:
## Step 1: Subset Relationship
Since $ B $ is a proper subset of $ C $, $ C $ must contain all elements of $ B $ and at least one additional element. Mathematically, $ n(C)>n(B) $.
## Step 2: Analyze the Maximum of $ n(C) $
There is no upper bound on the number of elements in $ C $ as long as $ n(C)>n(B) $ (here $ n(B) = 8 $). So $ C $ can have an arbitrarily large number of elements, meaning there is no limit on the number of elements in $ C $ (or equivalently, if we misinterpret the question as about $ B $, but since $ B $ is a proper subset of $ C $ and $ n(B) = 8 $, $ C $ can be as large as possible, so $ B $ (if the question was about $ B $) – no, the question is about $ C $. But from the options, option B says "There is no limit on how many elements B could have" – maybe a mislabeling, but the conclusion is that there's no maximum for $ n(C) $ (or for $ n(B) $ if the question was misphrased) because sets can be infinite or arbitrarily large finite sets.

# Answer for part b (assuming the question is about the maximum number of elements in $ C $ or $ B $ with the given option):
There is no limit (corresponding to option B: "There is no limit on how many elements B could have" – likely a misphrasing, but the idea is no upper bound).

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QUESTION IMAGE

Question

Explanation:

Part a: If \( n(C) = 8 \), what is the maximum number of elements in \( B \)?

Step 1: Recall proper subset definition

Step 2: Calculate maximum size of \( B \)

Step 1: Recall subset relationship

Step 2: Determine maximum size of \( C \)

Step 1: Proper Subset Definition

Step 2: Calculate the Maximum Size of \( B \)

Answer:

Part b: If \( n(B) = 8 \), what is the maximum number of elements in \( C \)?