Consider the set $\mathrm{N}$ of positive integers to be the universal set. Sets $\mathrm{H}, \mathrm{T}, \mathrm{E}$, and $\mathrm{P}$ are defined to the right. Determine whether or not the sets $E^{\prime}$ and $T^{\prime}$ are disjoint.
\[
\begin{array}{l}
H=\{n \in N \mid n> 150\} \\
T=\{n \in N \mid n< 1,000\} \\
E=\{n \in N \mid n \text { is even }\} \\
P=\{n \in N \mid n \text { is prime }\}
\end{array}
\]
Are $\mathrm{E}^{\prime}$ and $\mathrm{T}^{\prime}$ disjoint?
A. Yes, because there are no even numbers less than or equal to 1,000.
B. No, because there are no odd numbers greater than or equal to 1,000 .
C. Yes, because there is at least one even number that is less than or equal to 1,000 .
D. No, because there is at least one odd number that is greater than or equal to 1,000 .
Answer
Final Answer: \(\boxed{\text{D. No, because there is at least one odd number that is greater than or equal to 1,000}}\)
Step 1 ：The problem is asking whether the sets E' and T' are disjoint. E' is the complement of set E, which contains all even numbers, so E' contains all odd numbers. T' is the complement of set T, which contains all numbers less than 1000, so T' contains all numbers greater than or equal to 1000.
Step 2 ：We need to determine whether there is an odd number that is greater than or equal to 1000. If there is, then the sets E' and T' are not disjoint. If there isn't, then the sets E' and T' are disjoint.
Step 3 ：Since there are odd numbers that are greater than or equal to 1000 (for example, 1001), we can conclude that the sets E' and T' are not disjoint.
Step 4 ：Final Answer: \(\boxed{\text{D. No, because there is at least one odd number that is greater than or equal to 1,000}}\)
