Given $R$ and $T$ be sets defined as follows.
$R = \{2,4,6,7,8\}$
$T = \{x \in \mathbb{Z}:x \leq 4 \,\text{or}\, x \geq 8\}$
Question: Find the following
$\mathcal{P}(R)\cap \mathcal{P}(T)$
What is the solution to this? I'm confused about how to find the power set of $T$.
Best Answer
$\text{Hi Ben, hope this answer isn't overlong.}$
You have two sets, $R$, and $T$.
$R$ contains numbers $2,4,6,7,8$.
$T$ contains all integers less than or equal to $4$ and all integers greater than or equal to $8$. I.e., contains all integers, except $5, 6, 7$.
The expression $\mathcal P(R)\cap \mathcal P(T)$ represents the intersection of the power sets of $R$ and $T$. The power set of a set is the set of all of the set's subsets. That means it includes only values found in the original set. The intersection of two power sets therefore excludes all subsets in which values not common to the two original sets are found. So:
$\mathcal P(R)\cap \mathcal P(T) = \mathcal P(R\cap T)$
To compute $R\cap T$:
Notice that $R$ contains integers between $4$ and $8$--namely, $6,7$. But $T$ does not. Removing these, the intersection $R\cap T$ = $\{2,4,8\}$.
The answer is the power set of this intersection:
$\mathcal P(R)\cap \mathcal P(T)$
$ = \mathcal P(R\cap T)$
$ = \mathcal P(\{2,4,8\}) $
$= \boxed{\{\emptyset, \{2\}, \{4\}, \{8\}, \{2,4\}, \{4,8\}, \{2,8\}, \{2,4,8\}\}}$
Please reach out in the comments below if you are still uncertain as to how to solve this type of problem or if you are having additional questions or concerns.