Let's analyze the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as defined:
1. Set [tex]\( A \)[/tex] Definition:
- [tex]\( A \)[/tex] is the set of even numbers greater than or equal to 4. This means [tex]\( A \)[/tex] includes numbers such as 4, 6, 8, 10, and so forth, extending infinitely.
2. Set [tex]\( B \)[/tex] Definition:
- [tex]\( B = \{-30, -29, -28, -23, -22, 25\} \)[/tex]. This set contains six specific integers listed explicitly.
Now, let's determine the cardinalities and truth values of the statements:
### Cardinalities:
- Cardinality of [tex]\( A \)[/tex], denoted as [tex]\( n(A) \)[/tex]:
- Since [tex]\( A \)[/tex] is the set of even numbers starting from 4 and going on infinitely, [tex]\( n(A) \)[/tex] is infinite.
[tex]\[
n(A) = \text{infinite}
\][/tex]
- Cardinality of [tex]\( B \)[/tex], denoted as [tex]\( n(B) \)[/tex]:
- [tex]\( B \)[/tex] has 6 elements: [tex]\(-30, -29, -28, -23, -22, 25\)[/tex].
[tex]\[
n(B) = 6
\][/tex]
### Truth Values:
- Statement: [tex]\( 13 \in A \)[/tex]:
- 13 is not an even number, so it cannot be in [tex]\( A \)[/tex].
[tex]\[
13 \in A \quad \text{is} \quad \text{False}
\][/tex]
- Statement: [tex]\( 25 \in B \)[/tex]:
- The set [tex]\( B \)[/tex] contains the number 25.
[tex]\[
25 \in B \quad \text{is} \quad \text{True}
\][/tex]
- Statement: [tex]\( 6 \notin A \)[/tex]:
- 6 is an even number greater than or equal to 4, so it is in [tex]\( A \)[/tex].
[tex]\[
6 \notin A \quad \text{is} \quad \text{False}
\][/tex]
- Statement: [tex]\( -30 \in B \)[/tex]:
- The set [tex]\( B \)[/tex] contains the number -30.
[tex]\[
-30 \in B \quad \text{is} \quad \text{True}
\][/tex]
### Final Results:
[tex]\[
\begin{aligned}
n(A) &= \text{infinite} \\
n(B) &= 6 \\
\end{aligned}
\][/tex]
[tex]\[
\begin{tabular}{|r|c|c|}
\hline & \text{True} & \text{False} \\
\hline $13 \in A$ & & \bigcirc \\
\hline $25 \in B$ & \bigcirc & \\
\hline $6 \notin A$ & & \bigcirc \\
\hline $-30 \in B$ & \bigcirc & \\
\hline
\end{tabular}
\][/tex]
