16. Determine whether these biconditionals are true or

false.

a) 2 + 2 = 4 if and only if 1 + 1 = 2.

b) 1 + 1 = 2 if and only if 2 + 3 = 4.

c) 1 + 1 = 3 if and only if monkeys can fly.

d) 0 > 1 if and only if 2 > 1.

To determine whether these biconditionals are true or false, we need to analyze each statement individually. A biconditional statement "P if and only if Q" (denoted as \( P \leftrightarrow Q \)) is true if both \( P \) and \( Q \) are either both true or both false. If one is true and the other is false, the biconditional is false.

Let's evaluate each statement:

### a) \( 2 + 2 = 4 \) if and only if \( 1 + 1 = 2 \).
- \( 2 + 2 = 4 \) is **true**.
- \( 1 + 1 = 2 \) is **true**.
- Since both statements are true, the biconditional is **true**.

### b) \( 1 + 1 = 2 \) if and only if \( 2 + 3 = 4 \).
- \( 1 + 1 = 2 \) is **true**.
- \( 2 + 3 = 4 \) is **false**.
- Since one statement is true and the other is false, the biconditional is **false**.

### c) \( 1 + 1 = 3 \) if and only if monkeys can fly.
- \( 1 + 1 = 3 \) is **false**.
- The statement "monkeys can fly" is also **false** (assuming we're speaking literally).
- Since both statements are false, the biconditional is **true**.

### d) \( 0 > 1 \) if and only if \( 2 > 1 \).
- \( 0 > 1 \) is **false**.
- \( 2 > 1 \) is **true**.
- Since one statement is false and the other is true, the biconditional is **false**.

### Summary:
- a) True
- b) False
- c) True
- d) False

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What are the conditions for a biconditional statement to be true?
2. How does a biconditional differ from a regular conditional statement?
3. Can a biconditional be used to prove mathematical theorems? How?
4. What is the truth table for a biconditional statement?
5. How does the concept of biconditionals relate to logical equivalence?

**Tip:** In logic, a biconditional is only true when both components have the same truth value. This is crucial for understanding complex logical statements.

Explore the truth values of biconditional statements with detailed analysis and examples. Understand the conditions under which biconditionals are true or false. Suitable for advanced high school students and above.

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