Give the negation of each of the following statements.
Some students passed this course.

Answer

To find the negation of the statement "Some students passed this course," we can follow these steps:

### Step 1: Understand the Original Statement
The original statement "Some students passed this course" implies that at least one student has successfully passed the course.

### Step 2: Identify the Logical Structure
The phrase "some students" can be interpreted as "there exists at least one student" who passed the course. In logical terms, this can be represented as:
- ∃x (P(x)), where P(x) means "x passed the course."

### Step 3: Formulate the Negation
To negate the statement "Some students passed this course," we need to express that it is not true that at least one student passed. The logical negation of "there exists" (∃) is "for all" (∀). Therefore, the negation can be expressed as:
- ¬(∃x (P(x))) which translates to "For all students, it is not the case that they passed the course."

### Step 4: Rewrite the Negation
The negation can be rewritten in simpler terms. If no student passed the course, we can say:
- "No student passed this course."

### Final Answer
Thus, the negation of the statement "Some students passed this course" is:
- **No student passed this course.**

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