Let's break down the problem step by step.
### Part (a): List the Elements of Set [tex]\( B \)[/tex] and Set [tex]\( C \)[/tex]
1. Set [tex]\( A \)[/tex] is given as [tex]\(\{4, 5, 6\}\)[/tex].
2. Set [tex]\( B \)[/tex] is defined by the rule [tex]\( B = \{ y \colon y = 2x, \, x \in A \} \)[/tex].
- For [tex]\( x = 4 \)[/tex]: [tex]\( y = 2 \times 4 = 8 \)[/tex]
- For [tex]\( x = 5 \)[/tex]: [tex]\( y = 2 \times 5 = 10 \)[/tex]
- For [tex]\( x = 6 \)[/tex]: [tex]\( y = 2 \times 6 = 12 \)[/tex]
Therefore, the elements of set [tex]\( B \)[/tex] are [tex]\( \{8, 10, 12\} \)[/tex].
3. Set [tex]\( C \)[/tex] is defined as the first three multiples of 7.
- [tex]\( 7 \times 1 = 7 \)[/tex]
- [tex]\( 7 \times 2 = 14 \)[/tex]
- [tex]\( 7 \times 3 = 21 \)[/tex]
Therefore, the elements of set [tex]\( C \)[/tex] are [tex]\( \{7, 14, 21\} \)[/tex].
### Part (b): Show the Set [tex]\( A \)[/tex] and Set [tex]\( B \)[/tex] in a Venn Diagram
To represent this visually:
- Set [tex]\( A = \{4, 5, 6\} \)[/tex]
- Set [tex]\( B = \{8, 10, 12\} \)[/tex]
Since there are no common elements between sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we place them in separate circles that do not overlap:
```
[ A ] [ B ]
{4, 5, 6} {8, 10, 12}
```
Here, the circle on the left represents set [tex]\( A \)[/tex], and the circle on the right represents set [tex]\( B \)[/tex]. Since they have no elements in common, the circles do not intersect.
### Part (c): State Whether Set [tex]\( A \)[/tex] and [tex]\( C \)[/tex] are Overlapping or Disjoint Sets
- Set [tex]\( A = \{4, 5, 6\} \)[/tex]
- Set [tex]\( C = \{7, 14, 21\} \)[/tex]
To determine if sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex] are overlapping or disjoint, we check for any common elements.
- The elements of set [tex]\( A \)[/tex] are 4, 5, and 6.
- The elements of set [tex]\( C \)[/tex] are 7, 14, and 21.
There are no common elements between set [tex]\( A \)[/tex] and set [tex]\( C \)[/tex].
Hence, sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex] are disjoint.
Reason:
Two sets are disjoint if they have no elements in common. Since there are no shared elements between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], they are disjoint sets.
