**a.** Express the quantified statement in an equivalent​ way, that​ is, in a way that has exactly the same meaning.

**b.** Write the negation of the quantified statement.​ (The negation should begin with​ "all," "some," or​ "no.")

Some 

beetlesbeetles 

are not carnivorous.carnivorous.

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Part 1

**a.** Which of the following expresses the quantified statement in an equivalent​ way?

A.There are no beetles that are not carnivorous.

There are no beetles that are not carnivorous.There are no beetles that are not carnivorous.

B.All beetles are not carnivorous.

All beetles are not carnivorous.All beetles are not carnivorous.

C.Not all beetles are carnivorous.

Not all beetles are carnivorous.Not all beetles are carnivorous.

D.At least one beetle is carnivorous.

At least one beetle is carnivorous.At least one beetle is carnivorous.

Part 2

**b.** Which of the following is the negation of the quantified​ statement?

A.No beetles are carnivorous.

No beetles are carnivorous.No beetles are carnivorous.

B.Not all beetles are carnivorous.

Not all beetles are carnivorous.Not all beetles are carnivorous.

C.Some beetles are carnivorous.

Some beetles are carnivorous.Some beetles are carnivorous.

D.All beetles are carnivorous.

All beetles are carnivorous.

Let's go through the parts step by step.

### **Part 1: Equivalent Statement**

The given statement is:
**"Some beetles are not carnivorous."**

To express this statement in an equivalent way, consider the following options:

- **A.** There are no beetles that are not carnivorous.
- **B.** All beetles are not carnivorous.
- **C.** Not all beetles are carnivorous.
- **D.** At least one beetle is carnivorous.

**Analysis:**

- **A** says that no beetles are non-carnivorous, which contradicts the original statement.
- **B** says that every beetle is non-carnivorous, which also contradicts the original statement.
- **C** correctly reflects the idea that "some beetles are not carnivorous," meaning "not all beetles are carnivorous."
- **D** says at least one beetle is carnivorous, which is unrelated to the original statement.

**Correct Answer:** **C. Not all beetles are carnivorous.**

### **Part 2: Negation of the Statement**

The negation of the original statement **"Some beetles are not carnivorous"** should express that the original statement is false.

Consider the following options:

- **A.** No beetles are carnivorous.
- **B.** Not all beetles are carnivorous.
- **C.** Some beetles are carnivorous.
- **D.** All beetles are carnivorous.

**Analysis:**

- **A** suggests that all beetles are non-carnivorous, which is not a direct negation.
- **B** is actually equivalent to the original statement, so it's not a negation.
- **C** does not directly contradict the original statement.
- **D** is the correct negation. If "some beetles are not carnivorous" is false, then it must be true that "all beetles are carnivorous."

**Correct Answer:** **D. All beetles are carnivorous.**

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Would you like more details or have any questions?

Here are some related questions you might find useful:
1. What is the negation of "All swans are white"?
2. How can you express "Some students are not athletes" in an equivalent way?
3. What is the equivalent statement for "No dogs are vegetarian"?
4. How do you negate the statement "Some apples are red"?
5. What is the equivalent statement for "Not all books are fiction"?
6. What is the negation of "At least one car is electric"?
7. How do you express "Some flowers are not roses" in an equivalent way?
8. How do you negate "Every human has a heart"?

**Tip:** When dealing with quantified statements, the negation often involves flipping "some" to "all" or vice versa.

Learn how to express and negate quantified statements effectively with examples. Explore equivalent statements and their negations, crucial for understanding logic in high school mathematics.

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