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Let's break down the problem step by step as requested: --- ### Step 1: **Understanding the Problem** We are given: \[ \bigcap_{i=1}^n \{X_i \text{ is } T_2\} \] And it is stated that: \[ \bigcap_{i=1}^n \{X_i \text{ is } T_2\} \iff X_i \text{ is } T_2 \text{ for all } i \in \{1,...,n\} \] **Explanation:** The intersection notation means "all of the $X_i$'s are $T_2$." The $\iff$ (if and only if) tells us that the statement is true exactly when *each* $X_i$ is $T_2$ for every $i$ in the set $\{1, ..., n\}$. --- ### Step 2: **Breaking Down the Logical Equivalence** - $\bigcap_{i=1}^n \{X_i \text{ is } T_2\}$ means "the set of all outcomes where every $X_i$ is $T_2$." - "$X_i$ is $T_2$ for all $i$" is simply the condition that each and every $X_i$ satisfies the $T_2$ property. **Explanation:** This step confirms that the intersection of these sets is logically equivalent to saying that each element in the index set has the $T_2$ property. --- ### Step 3: **Final Statement** So, the statement translates to: \[ \bigcap_{i=1}^n \{X_i \text{ is } T_2\} \iff \text{all } X_i \text{ are } T_2 \text{ for } i=1,2,\ldots,n \] **Explanation:** This means that the intersection of sets where $X_i$ is $T_2$ for $i=1$ to $n$ is true only if every $X_i$ individually is $T_2$. --- ## **Final Answer:** \[ \boxed{ \bigcap_{i=1}^n \{X_i \text{ is } T_2\} \iff X_i \text{ is } T_2 \text{ for all } i \in \{1, ..., n\} } \] This simply means "the intersection of the $T_2$ property for each $X_i$ holds if and only if every $X_i$ satisfies the $T_2$ property individually."