Let's analyze the given set and identify the values of the maximum, supremum, minimum, and infimum.

Given Set Definition:

The set $E$ is defined as:

$E = \left\{ 1 - \sum_{m=1}^{14} \frac{m}{n} : n \in \mathbb{N} \right\}$

Breaking Down the Expression:

1. Summation: $\sum_{m=1}^{14} \frac{m}{n}$ is a summation of a series from $m = 1$ to $m = 14$. This equals:

$\sum_{m=1}^{14} \frac{m}{n} = \frac{1 + 2 + 3 + \ldots + 14}{n} = \frac{\frac{14 \cdot 15}{2}}{n} = \frac{105}{n}$

2. Set $E$: Each element in the set is of the form:

$1 - \frac{105}{n}$

where $n \in \mathbb{N}$. As $n$ takes different values, $E$ will vary based on $n$.

Evaluating $E$:

• As $n$ becomes larger, $\frac{105}{n}$ approaches 0, making $1 - \frac{105}{n}$ approach 1.
• For $n = 1$, $1 - \frac{105}{1} = 1 - 105 = -104$.
• For $n = 2$, $1 - \frac{105}{2} = 1 - 52.5 = -51.5$.
• As $n$ increases, the values approach 1 from the left.

Analyzing Extremes:

1. Supremum ($\sup E$) is the least upper bound of the set. Here, the supremum is 1.
2. Maximum ($\max E$) is the greatest element in the set, if it exists. As $n$ becomes large, $1 - \frac{105}{n} \approx 1$, but 1 is not achieved for any finite $n$. Hence, $\max E$ does not exist.
3. Infimum ($\inf E$) is the greatest lower bound. Since $n = 1$ gives $-104$, which is the smallest value, the infimum is $-104$.
4. Minimum ($\min E$) is the least element in the set. It exists and is $-104$ when $n = 1$.

Conclusion:

Let's match this with the provided options:

• Option A: $\max E = 1, \sup E = 1, \min E = 23, \inf E = 23$. This is incorrect.
• Option B: $\max E = \frac{3}{4}, \sup E = \frac{3}{4}, \min E = \frac{2}{3}, \inf E$ does not exist. This is incorrect.
• Option C: (\max E = \frac{3}{4}, \sup E