Let $A = (-2, +\infty)$ and $B = [5, +\infty)$. Find the complement of $B$ on $A$, which can be denoted as $A \setminus B$ or $A - B$.

1. Problem Description

Let $A = (-2, +\infty)$ and $B = [5, +\infty)$. Find the complement of $B$ on $A$, which can be denoted as $A \setminus B$ or $A - B$.

2. Solution Steps

The complement of $B$ on $A$, denoted as $A \setminus B$, is the set of all elements in $A$ that are not in $B$. In other words, $A \setminus B = \{x \mid x \in A \text{ and } x \notin B\}$.

We have $A = (-2, +\infty) = \{x \mid -2 < x < +\infty\}$ and $B = [5, +\infty) = \{x \mid 5 \le x < +\infty\}$.

We want to find $A \setminus B$, which consists of elements in $A$ but not in $B$.

$A \setminus B = (-2, +\infty) \setminus [5, +\infty)$. We can express $A \setminus B$ as an intersection:

$A \setminus B = A \cap B^c$, where $B^c$ is the complement of $B$. The complement of $B = [5, +\infty)$ is $B^c = (-\infty, 5)$.

So, $A \setminus B = (-2, +\infty) \cap (-\infty, 5) = \{x \mid -2 < x < +\infty \text{ and } -\infty < x < 5\}$.

This intersection is the set of $x$ such that $-2 < x < 5$. Thus, $A \setminus B = (-2, 5)$.

However, none of the answer choices match $(-2, 5)$. Let's re-examine the original text. It states "the complement of A on U". There appears to be a typo. The problem likely intended to ask for the complement of B with respect to A, or the complement of the union of A and B.

If it asked for $A \cup B$, we would have $A \cup B = (-2, +\infty) \cup [5, +\infty) = (-2, +\infty)$. If we're considering the universal set to be the real numbers $\mathbb{R}$, then the complement of this is $(-\infty, -2]$.

If instead it asked for $A \cap B$, then $A \cap B = (-2, +\infty) \cap [5, +\infty) = [5, +\infty)$. Then the complement would be $(-\infty, 5)$.

If the question intended to define U as the union of A and B, denoted as $U = A \cup B$, and asked to compute the complement of $B$ on $U$, that means we want to find $U \setminus B$.

Since $A = (-2, +\infty)$ and $B = [5, +\infty)$, $U = A \cup B = (-2, +\infty)$. Then $U \setminus B = (-2, +\infty) \setminus [5, +\infty) = (-2, 5)$.

Then we might be looking for $[-2, 5]$ rather than $(-2, 5)$.

If the question is asking for the complement of the intersection of A and B with respect to the universal set $U = \mathbb{R}$, then we have $A \cap B = [5, +\infty)$. The complement of this is $(-\infty, 5)$.

Consider the option [A] [-2, 5]. This may refer to the interval $(-2, 5]$. We found that $A \setminus B = (-2, 5)$. It seems like the closest answer would be (-2, 5), where the parenthesis means that -2 and 5 are not included. However, there may be a typo and [A] is meant to be (-2, 5].

With $A = (-2, \infty)$ and $B=[5,\infty)$, the complement of B with respect to A is the set of all elements in A that are not in B, or $A-B$. This gives us the interval $(-2,5)$. It seems closest to answer choice C (-2,5).

3. Final Answer

C (-2,5)