U= all natural numbers less than or equal to 20

Question

A= factors of 12
B= even numbers greater than 2 but less than or equal to 10
C= factors of 16
List the members of:
I. A intersection B intersection C
Ii. A intersection B' intersection C
Iii. A union B intersection C'
IV. A' intersection B intersection C
V. A' intersection B' intersection C
Vi. A' intersection B intersection C'
Vii. A intersection B' intersection C'
Viii. (A union B union C)'

Answer 1

Let's start by identifying the sets based on the given definitions.

Universal Set (U): All natural numbers less than or equal to 20
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Set A (factors of 12): The factors of 12 are {1, 2, 3, 4, 6, 12}
A = {1, 2, 3, 4, 6, 12}

Set B (even numbers greater than 2 but less than or equal to 10): The even numbers in this range are {4, 6, 8, 10}
B = {4, 6, 8, 10}

Set C (factors of 16): The factors of 16 are {1, 2, 4, 8, 16}
C = {1, 2, 4, 8, 16}

Now let’s calculate the required set operations.

I. A ∩ B ∩ C

We find the intersection of sets A, B, and C:
A ∩ B = {4, 6}
B ∩ C = {4, 8}
A ∩ C = {4}

Combining these, we find:
A ∩ B ∩ C = {4}

II. A ∩ B' ∩ C

First, we find B':
B' = U - B = {1, 2, 3, 5, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Now we find the intersection:
A ∩ B' = {1, 2, 3}
A ∩ B' ∩ C = {1, 2, 3} ∩ {1, 2, 4, 8, 16}
This gives us:
A ∩ B' ∩ C = {1, 2}

III. A ∪ B ∩ C'

First, we find C':
C' = U - C = {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}

Then we find the union:
A ∪ B = {1, 2, 3, 4, 6, 12} ∪ {4, 6, 8, 10} = {1, 2, 3, 4, 6, 8, 10, 12}

Now we intersect this with C':
(A ∪ B) ∩ C' = {1, 2, 3, 4, 6, 8, 10, 12} ∩ {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}
This gives us:
(A ∪ B) ∩ C' = {6, 10, 12}

IV. A' ∩ B ∩ C

We first find A':
A' = U - A = {5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20}

Now we can find:
A' ∩ B = {8, 10}
Now we intersect that with C:
(A' ∩ B) ∩ C = {8, 10} ∩ {1, 2, 4, 8, 16}
This gives us:
(A' ∩ B) ∩ C = {8}

V. A' ∩ B' ∩ C

We already have A' and B': A' = {5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20}
B' = {1, 2, 3, 5, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Now we find the intersection with C:
A' ∩ B' = {5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 20}
Now we find:
(A' ∩ B') ∩ C = {5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 20} ∩ {1, 2, 4, 8, 16} = {}
Thus, we have:
A' ∩ B' ∩ C = {}

VI. A' ∩ B ∩ C'

We have A', B and C': A' = {5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20}
B = {4, 6, 8, 10}
C' = {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}

Now we find the intersection:
A' ∩ B = {8, 10}
Then intersect that with C':
(A' ∩ B) ∩ C' = {8, 10} ∩ {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20} = {10}
Thus:
A' ∩ B ∩ C' = {10}

VII. A ∩ B' ∩ C'

From earlier:
A ∩ B' = {1, 2, 3}
Now we intersect that with C':
A ∩ B' ∩ C' = {1, 2, 3} ∩ {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20} = {3}
Thus:
A ∩ B' ∩ C' = {3}

VIII. (A ∪ B ∪ C)'

Now we can find the union of A, B, and C:
A ∪ B ∪ C = {1, 2, 3, 4, 6, 8, 10, 12, 16}

Finally, we subtract from U to find:
(A ∪ B ∪ C)' = U - (A ∪ B ∪ C) = {5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 20}

In summary, we have:

1. I. A ∩ B ∩ C = {4}
2. II. A ∩ B' ∩ C = {1, 2}
3. III. A ∪ B ∩ C' = {6, 10, 12}
4. IV. A' ∩ B ∩ C = {8}
5. V. A' ∩ B' ∩ C = {}
6. VI. A' ∩ B ∩ C' = {10}
7. VII. A ∩ B' ∩ C' = {3}
8. VIII. (A ∪ B ∪ C)' = {5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 20}