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]
