To solve this problem, we'll begin by defining the sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] and then proceed to find the intersection and union of these sets. Finally, we will find the required values for the operations given.
### 1. Define the Sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
- Set [tex]\(A\)[/tex]: The set of multiples of 4 less than 40.
[tex]\(A = \{4, 8, 12, 16, 20, 24, 28, 32, 36\}\)[/tex]
- Set [tex]\(B\)[/tex]: The set of multiples of 6 less than 40.
[tex]\(B = \{6, 12, 18, 24, 30, 36\}\)[/tex]
### 2. Venn Diagram
Let's first understand how sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] overlap. The multiples of both 4 and 6 that are less than 40 will be in the intersection.
We have:
- Common multiples of 4 and 6 (i.e., multiples of 12) less than 40: [tex]\(12, 24, 36\)[/tex]
So, the intersection [tex]\(A \cap B\)[/tex] is:
[tex]\(A \cap B = \{12, 24, 36\}\)[/tex]
Now, let's draw the Venn diagram:
```
________ ________
| | | |
| A | | B |
|{4, 8, | {6, 18,|30 |
| 28, | 12,24, | 36} |
|32, | 36} | |
|________| |________|
```
### 3. Calculations
Intersection [tex]\(A \cap B\)[/tex]:
- [tex]\(A \cap B = \{12, 24, 36\}\)[/tex]
- The number of elements in the intersection ([tex]\(n(A \cap B)\)[/tex]):
[tex]\(n(A \cap B) = 3\)[/tex]
Union [tex]\(A \cup B\)[/tex]:
- Combine all unique elements from sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\(A \cup B = \{4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36\}\)[/tex]
- The number of unique elements in the union ([tex]\(n(A \cup B)\)[/tex]):
[tex]\(n(A \cup B) = 12\)[/tex]
Difference [tex]\((A \cup B) - (A \cap B)\)[/tex]:
- The elements in the union that are not in the intersection:
[tex]\((A \cup B) - (A \cap B) = \{4, 6, 8, 16, 18, 20, 28, 30, 32\}\)[/tex]
- The number of elements in this difference ([tex]\(n[(A \cup B) - (A \cap B)]\)[/tex]):
[tex]\(n[(A \cup B) - (A \cap B)] = 9\)[/tex]
### Summary
- Sets:
- [tex]\(A = \{4, 8, 12, 16, 20, 24, 28, 32, 36\}\)[/tex]
- [tex]\(B = \{6, 12, 18, 24, 30, 36\}\)[/tex]
- Venn Diagram Representation:
```
________ ________
| | | |
| A | | B |
|{4, 8, | {6, 18,|30 |
| 28, | 12,24, | 36} |
|32, | 36} | |
|________| |________|
```
- Operations:
a) [tex]\(n(A \cap B) = 3\)[/tex]
b) [tex]\(n[(A \cup B) - (A \cap B)] = 9\)[/tex]
