Question

U= all natural numbers less than or equal to 20
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}**