Answer :
We are given the universal set
[tex]$$
U = \{2,\ 4,\ 6,\ 9,\ 10,\ 14,\ 15,\ 16,\ 18\}.
$$[/tex]
We define the subsets as follows:
- Subset [tex]$A$[/tex] (even numbers):
[tex]$$A = \{2, 4, 6, 10, 14, 16, 18\}.$$[/tex]
- Subset [tex]$B$[/tex] (square numbers):
Since in [tex]$U$[/tex], the numbers that are perfect squares are [tex]$4$[/tex] (since [tex]$2^2=4$[/tex]), [tex]$9$[/tex] (since [tex]$3^2=9$[/tex]), and [tex]$16$[/tex] (since [tex]$4^2=16$[/tex]), we have
[tex]$$B = \{4, 9, 16\}.$$[/tex]
### 10.1.1 Representation by Venn Diagram
Draw two overlapping circles, one for [tex]$A$[/tex] and one for [tex]$B$[/tex], inside a rectangle representing [tex]$U$[/tex].
1. Intersection ([tex]$A\cap B$[/tex]):
The numbers that belong to both [tex]$A$[/tex] and [tex]$B$[/tex] are
[tex]$$A\cap B = \{4, 16\}.$$[/tex]
2. Only in [tex]$A$[/tex] (i.e., [tex]$A-B$[/tex]):
Remove the elements in [tex]$A\cap B$[/tex] from [tex]$A$[/tex]:
[tex]$$A - B = \{2, 6, 10, 14, 18\}.$$[/tex]
3. Only in [tex]$B$[/tex] (i.e., [tex]$B-A$[/tex]):
Remove the elements in [tex]$A\cap B$[/tex] from [tex]$B$[/tex]:
[tex]$$B - A = \{9\}.$$[/tex]
4. Outside both [tex]$A$[/tex] and [tex]$B$[/tex]:
These are the elements in [tex]$U$[/tex] that are neither even nor square:
[tex]$$U - (A \cup B) = \{15\}.$$[/tex]
Your Venn diagram should show:
- Left circle ([tex]$A$[/tex]): contains [tex]$\{2, 6, 10, 14, 18\}$[/tex].
- Right circle ([tex]$B$[/tex]): contains [tex]$\{9\}$[/tex].
- Overlap: contains [tex]$\{4, 16\}$[/tex].
- Outside both circles (but inside [tex]$U$[/tex]): contains [tex]$\{15\}$[/tex].
### 10.1.2 Determine [tex]$P(A \text{ or } B)$[/tex]
The probability of an element being in [tex]$A$[/tex] or [tex]$B$[/tex] (or both) is given by the ratio of the number of elements in the union of [tex]$A$[/tex] and [tex]$B$[/tex] to the total number of elements in [tex]$U$[/tex].
First, find the union:
[tex]$$
A \cup B = \{2, 4, 6, 9, 10, 14, 16, 18\}.
$$[/tex]
There are [tex]$8$[/tex] elements in [tex]$A \cup B$[/tex], and the total number of elements in [tex]$U$[/tex] is [tex]$9$[/tex]. Thus,
[tex]$$
P(A \cup B) = \frac{8}{9} \approx 0.8889.
$$[/tex]
### 10.1.3 Determine [tex]$P(A \text{ and } B)$[/tex]
The probability of an element being in both [tex]$A$[/tex] and [tex]$B$[/tex] is given by the ratio of the number of elements in the intersection [tex]$A\cap B$[/tex] to the total number of elements in [tex]$U$[/tex].
Since
[tex]$$
A \cap B = \{4, 16\},
$$[/tex]
there are [tex]$2$[/tex] elements in the intersection. Therefore,
[tex]$$
P(A \cap B) = \frac{2}{9} \approx 0.2222.
$$[/tex]
### 10.1.4 Determine [tex]$P\left(A' \text{ and } B'\right)$[/tex]
The probability of an element being in neither [tex]$A$[/tex] nor [tex]$B$[/tex] is found by considering the complement of [tex]$A \cup B$[/tex] in [tex]$U$[/tex]. We already determined that the elements outside both [tex]$A$[/tex] and [tex]$B$[/tex] are:
[tex]$$
U - (A \cup B) = \{15\}.
$$[/tex]
There is [tex]$1$[/tex] element in this region. Hence,
[tex]$$
P\left(A' \cap B'\right) = \frac{1}{9} \approx 0.1111.
$$[/tex]
### Final Answers Summary
- Venn Diagram Representation:
- [tex]$A$[/tex] only: [tex]$\{2, 6, 10, 14, 18\}$[/tex]
- [tex]$B$[/tex] only: [tex]$\{9\}$[/tex]
- [tex]$A \cap B$[/tex]: [tex]$\{4, 16\}$[/tex]
- Outside both: [tex]$\{15\}$[/tex]
- Probability Calculations:
[tex]$$P(A \cup B) = \frac{8}{9} \quad \text{(approximately 0.8889)}$$[/tex]
[tex]$$P(A \cap B) = \frac{2}{9} \quad \text{(approximately 0.2222)}$$[/tex]
[tex]$$P\left(A' \cap B'\right) = \frac{1}{9} \quad \text{(approximately 0.1111)}$$[/tex]
[tex]$$
U = \{2,\ 4,\ 6,\ 9,\ 10,\ 14,\ 15,\ 16,\ 18\}.
$$[/tex]
We define the subsets as follows:
- Subset [tex]$A$[/tex] (even numbers):
[tex]$$A = \{2, 4, 6, 10, 14, 16, 18\}.$$[/tex]
- Subset [tex]$B$[/tex] (square numbers):
Since in [tex]$U$[/tex], the numbers that are perfect squares are [tex]$4$[/tex] (since [tex]$2^2=4$[/tex]), [tex]$9$[/tex] (since [tex]$3^2=9$[/tex]), and [tex]$16$[/tex] (since [tex]$4^2=16$[/tex]), we have
[tex]$$B = \{4, 9, 16\}.$$[/tex]
### 10.1.1 Representation by Venn Diagram
Draw two overlapping circles, one for [tex]$A$[/tex] and one for [tex]$B$[/tex], inside a rectangle representing [tex]$U$[/tex].
1. Intersection ([tex]$A\cap B$[/tex]):
The numbers that belong to both [tex]$A$[/tex] and [tex]$B$[/tex] are
[tex]$$A\cap B = \{4, 16\}.$$[/tex]
2. Only in [tex]$A$[/tex] (i.e., [tex]$A-B$[/tex]):
Remove the elements in [tex]$A\cap B$[/tex] from [tex]$A$[/tex]:
[tex]$$A - B = \{2, 6, 10, 14, 18\}.$$[/tex]
3. Only in [tex]$B$[/tex] (i.e., [tex]$B-A$[/tex]):
Remove the elements in [tex]$A\cap B$[/tex] from [tex]$B$[/tex]:
[tex]$$B - A = \{9\}.$$[/tex]
4. Outside both [tex]$A$[/tex] and [tex]$B$[/tex]:
These are the elements in [tex]$U$[/tex] that are neither even nor square:
[tex]$$U - (A \cup B) = \{15\}.$$[/tex]
Your Venn diagram should show:
- Left circle ([tex]$A$[/tex]): contains [tex]$\{2, 6, 10, 14, 18\}$[/tex].
- Right circle ([tex]$B$[/tex]): contains [tex]$\{9\}$[/tex].
- Overlap: contains [tex]$\{4, 16\}$[/tex].
- Outside both circles (but inside [tex]$U$[/tex]): contains [tex]$\{15\}$[/tex].
### 10.1.2 Determine [tex]$P(A \text{ or } B)$[/tex]
The probability of an element being in [tex]$A$[/tex] or [tex]$B$[/tex] (or both) is given by the ratio of the number of elements in the union of [tex]$A$[/tex] and [tex]$B$[/tex] to the total number of elements in [tex]$U$[/tex].
First, find the union:
[tex]$$
A \cup B = \{2, 4, 6, 9, 10, 14, 16, 18\}.
$$[/tex]
There are [tex]$8$[/tex] elements in [tex]$A \cup B$[/tex], and the total number of elements in [tex]$U$[/tex] is [tex]$9$[/tex]. Thus,
[tex]$$
P(A \cup B) = \frac{8}{9} \approx 0.8889.
$$[/tex]
### 10.1.3 Determine [tex]$P(A \text{ and } B)$[/tex]
The probability of an element being in both [tex]$A$[/tex] and [tex]$B$[/tex] is given by the ratio of the number of elements in the intersection [tex]$A\cap B$[/tex] to the total number of elements in [tex]$U$[/tex].
Since
[tex]$$
A \cap B = \{4, 16\},
$$[/tex]
there are [tex]$2$[/tex] elements in the intersection. Therefore,
[tex]$$
P(A \cap B) = \frac{2}{9} \approx 0.2222.
$$[/tex]
### 10.1.4 Determine [tex]$P\left(A' \text{ and } B'\right)$[/tex]
The probability of an element being in neither [tex]$A$[/tex] nor [tex]$B$[/tex] is found by considering the complement of [tex]$A \cup B$[/tex] in [tex]$U$[/tex]. We already determined that the elements outside both [tex]$A$[/tex] and [tex]$B$[/tex] are:
[tex]$$
U - (A \cup B) = \{15\}.
$$[/tex]
There is [tex]$1$[/tex] element in this region. Hence,
[tex]$$
P\left(A' \cap B'\right) = \frac{1}{9} \approx 0.1111.
$$[/tex]
### Final Answers Summary
- Venn Diagram Representation:
- [tex]$A$[/tex] only: [tex]$\{2, 6, 10, 14, 18\}$[/tex]
- [tex]$B$[/tex] only: [tex]$\{9\}$[/tex]
- [tex]$A \cap B$[/tex]: [tex]$\{4, 16\}$[/tex]
- Outside both: [tex]$\{15\}$[/tex]
- Probability Calculations:
[tex]$$P(A \cup B) = \frac{8}{9} \quad \text{(approximately 0.8889)}$$[/tex]
[tex]$$P(A \cap B) = \frac{2}{9} \quad \text{(approximately 0.2222)}$$[/tex]
[tex]$$P\left(A' \cap B'\right) = \frac{1}{9} \quad \text{(approximately 0.1111)}$$[/tex]
