To find the union and intersection of two sets [tex]\( B \)[/tex] and [tex]\( C \)[/tex] given by the following conditions:
[tex]\[
\begin{array}{l}
B = \{x \mid x > 2\} \\
C = \{x \mid x \geq 7\}
\end{array}
\][/tex]
we'll analyze each condition step-by-step.
### Step-by-step Solution:
1. Define [tex]\( B \)[/tex] in Interval Notation:
The set [tex]\( B \)[/tex] is defined as all [tex]\( x \)[/tex] such that [tex]\( x > 2 \)[/tex]. In interval notation, this is represented as:
[tex]\[
B = (2, \infty)
\][/tex]
2. Define [tex]\( C \)[/tex] in Interval Notation:
The set [tex]\( C \)[/tex] is defined as all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex]. In interval notation, this is represented as:
[tex]\[
C = [7, \infty)
\][/tex]
3. Union [tex]\( B \cup C \)[/tex]:
The union of two sets [tex]\( B \cup C \)[/tex] includes all elements that belong to either [tex]\( B \)[/tex] or [tex]\( C \)[/tex] or both.
Since [tex]\( B = (2, \infty) \)[/tex] includes all real numbers greater than 2 and [tex]\( C = [7, \infty) \)[/tex] includes all real numbers greater than or equal to 7, the union will be:
[tex]\[
B \cup C = (2, \infty)
\][/tex]
This is because [tex]\( (2, \infty) \)[/tex] already encompasses all elements of [tex]\( [7, \infty) \)[/tex].
4. Intersection [tex]\( B \cap C \)[/tex]:
The intersection of two sets [tex]\( B \cap C \)[/tex] includes all elements that are common to both [tex]\( B \)[/tex] and [tex]\( C \)[/tex].
Since [tex]\( B = (2, \infty) \)[/tex] includes all real numbers greater than 2 and [tex]\( C = [7, \infty) \)[/tex] includes all real numbers greater than or equal to 7, the intersection will be:
[tex]\[
B \cap C = [7, \infty)
\][/tex]
This is because the smallest number in [tex]\( [7, \infty) \)[/tex] that is also greater than 2 is 7, and all numbers greater than or equal to 7 are common to both sets.
Therefore, the interval notations for the union and intersection of [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are:
- [tex]\( B \cup C = (2, \infty) \)[/tex]
- [tex]\( B \cap C = [7, \infty) \)[/tex]
