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Let [tex]\( S \)[/tex] be the universal set, where:
[tex]\[ S=\{1,2,3, \ldots, 23,24,25\} \][/tex]

Let sets [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] be subsets of [tex]\( S \)[/tex]:

[tex]\[ A=\{2,12,13,15,16,17,19,20,22,23,24\} \][/tex]
[tex]\[ B=\{1,3,4,5,8,13,19,21\} \][/tex]
[tex]\[ C=\{3,4,7,10,11,13,16,18,19,22,24,25\} \][/tex]

1. Find the number of elements in the set [tex]\( A \cap B \)[/tex]:
[tex]\[ n(A \cap B) = \, \square \][/tex]

2. Find the number of elements in the set [tex]\( B \cap C \)[/tex]:
[tex]\[ n(B \cap C) = \, \square \][/tex]

3. Find the number of elements in the set [tex]\( A \cap C \)[/tex]:
[tex]\[ n(A \cap C) = \, \square \][/tex]

Sagot :

GinnyAnsweridn
To find the number of elements in the intersections of the sets [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], let's determine the elements common to each pair of sets and then count those elements.

### Finding [tex]\( A \cap B \)[/tex]:
We need to find the elements that are common to both [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. List the elements of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:

- [tex]\( A = \{2, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24\} \)[/tex]
- [tex]\( B = \{1, 3, 4, 5, 8, 13, 19, 21\} \)[/tex]

Identify the common elements:

[tex]\[ A \cap B = \{13, 19\} \][/tex]

Count the number of elements in [tex]\( A \cap B \)[/tex]:

[tex]\[ n(A \cap B) = 2 \][/tex]

### Finding [tex]\( B \cap C \)[/tex]:
We need to find the elements that are common to both [tex]\(B\)[/tex] and [tex]\(C\)[/tex]. List the elements of [tex]\(B\)[/tex] and [tex]\(C\)[/tex]:

- [tex]\( B = \{1, 3, 4, 5, 8, 13, 19, 21\} \)[/tex]
- [tex]\( C = \{3, 4, 7, 10, 11, 13, 16, 18, 19, 22, 24, 25\} \)[/tex]

Identify the common elements:

[tex]\[ B \cap C = \{3, 4, 13, 19\} \][/tex]

Count the number of elements in [tex]\( B \cap C \)[/tex]:

[tex]\[ n(B \cap C) = 4 \][/tex]

### Finding [tex]\( A \cap C \)[/tex]:
We need to find the elements that are common to both [tex]\(A\)[/tex] and [tex]\(C\)[/tex]. List the elements of [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:

- [tex]\( A = \{2, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24\} \)[/tex]
- [tex]\( C = \{3, 4, 7, 10, 11, 13, 16, 18, 19, 22, 24, 25\} \)[/tex]

Identify the common elements:

[tex]\[ A \cap C = \{13, 16, 19, 22, 24\} \][/tex]

Count the number of elements in [tex]\( A \cap C \)[/tex]:

[tex]\[ n(A \cap C) = 5 \][/tex]

### Final Results:
- [tex]\( n(A \cap B) = 2 \)[/tex]
- [tex]\( n(B \cap C) = 4 \)[/tex]
- [tex]\( n(A \cap C) = 5 \)[/tex]

Therefore:
[tex]\[ n(A \cap B) = 2 \][/tex]
[tex]\[ n(B \cap C) = 4 \][/tex]
[tex]\[ n(A \cap C) = 5 \][/tex]
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