Let's go through each part step by step.

(a) Determine 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶):

The intersection of sets 𝐴, 𝐵, and 𝐶, denoted $A \cap B \cap C$, consists of elements that are common to all three sets.

• 𝐴 = {𝑎, 𝑐, 𝑑, 𝑒, 𝑔, ℎ, 𝑚}
• 𝐵 = {𝑐, 𝑒, 𝑓, 𝑔, ℎ}
• 𝐶 = {𝑎, 𝑏, 𝑒, 𝑔, ℎ, 𝑘, 𝑚}

Let's find the common elements in all three sets.

• 𝑎 is in 𝐴, but not in 𝐵 or 𝐶.
• 𝑐 is in 𝐴 and 𝐵, but not in 𝐶.
• 𝑑 is only in 𝐴.
• 𝑒 is in 𝐴, 𝐵, and 𝐶.
• 𝑔 is in 𝐴, 𝐵, and 𝐶.
• ℎ is in 𝐴, 𝐵, and 𝐶.
• 𝑚 is in 𝐴 and 𝐶, but not in 𝐵.
• 𝑓 is only in 𝐵.
• 𝑏 is only in 𝐶.
• 𝑘 is only in 𝐶.

The common elements across all three sets are: {𝑒, 𝑔, ℎ}.

Thus, $A \cap B \cap C = \{𝑒, 𝑔, ℎ\}$, and the number of elements in this intersection is: $n(A \cap B \cap C) = 3.$

(b) Determine 𝑛(𝐴 ∩ 𝐵):

Now, find the intersection of sets 𝐴 and 𝐵, denoted $A \cap B$.

• 𝐴 = {𝑎, 𝑐, 𝑑, 𝑒, 𝑔, ℎ, 𝑚}
• 𝐵 = {𝑐, 𝑒, 𝑓, 𝑔, ℎ}

The common elements between 𝐴 and 𝐵 are: {𝑐, 𝑒, 𝑔, ℎ}.

Thus, $A \cap B = \{𝑐, 𝑒, 𝑔, ℎ\}$, and the number of elements in this intersection is: $n(A \cap B) = 4.$

(c) Determine 𝑛(𝐴 ∩ 𝐶):

Now, find the intersection of sets 𝐴 and 𝐶, denoted $A \cap C$.

• 𝐴 = {𝑎, 𝑐, 𝑑, 𝑒, 𝑔, ℎ, 𝑚}
• 𝐶 = {𝑎, 𝑏, 𝑒, 𝑔, ℎ, 𝑘, 𝑚}

The common elements between 𝐴 and 𝐶 are: {𝑎, 𝑒, 𝑔, ℎ, 𝑚}.

Thus, $A \cap C = \{𝑎, 𝑒, 𝑔, ℎ, 𝑚\}$, and the number of elements in this intersection is: $n(A \cap C) = 5.$

(d) Determine 𝑛(𝐵 ∩ 𝐶):

Next, find the intersection of sets 𝐵 and 𝐶, denoted $B \cap C$.

• 𝐵 = {𝑐, 𝑒, 𝑓, 𝑔, ℎ}
• 𝐶 = {𝑎, 𝑏, 𝑒, 𝑔, ℎ, 𝑘, 𝑚}

The common elements between 𝐵 and 𝐶 are: {𝑒, 𝑔, ℎ}.

Thus, $B \cap C = \{𝑒, 𝑔, ℎ\}$, and the number of elements in this intersection is: $n(B \cap C) = 3.$

(e) Determine 𝑛(𝐴):

The number of elements in set 𝐴 is simply the number of distinct elements in the set.

• 𝐴 = {𝑎, 𝑐, 𝑑, 𝑒, 𝑔, ℎ, 𝑚}

Thus, the number of elements in 𝐴 is: $n(A) = 7.$

(f) Determine 𝑛(𝐵):

The number of elements in set 𝐵 is:

• 𝐵 = {𝑐, 𝑒, 𝑓, 𝑔, ℎ}

Thus, the number of elements in 𝐵 is: $n(B) = 5.$

(g) Determine 𝑛(𝐶):

The number of elements in set 𝐶 is:

• 𝐶 = {𝑎, 𝑏, 𝑒, 𝑔, ℎ, 𝑘, 𝑚}

Thus, the number of elements in 𝐶 is: $n(C) = 7.$

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Summary of answers:

(a) $n(A \cap B \cap C) = 3$

(b) $n(A \cap B) = 4$

(c) $n(A \cap C) = 5$

(d) $n(B \cap C) = 3$

(e) $n(A) = 7$

(f) $n(B) = 5$

(g) $n(C) = 7$

Would you like further details on any part, or have more questions? Here's a list of related questions:

1. How do we compute the union of two or more sets?
2. What is the difference between intersection and union of sets?
3. How can we apply these concepts in probability theory?
4. What is a Venn diagram, and how can it help visualize these operations?
5. What are some real-world applications of set theory?

Tip: Understanding how to identify elements in set intersections can be a crucial skill in data analysis and problem-solving in general!