18. Determine whether each of these conditional statements

is true or false.

a) If 1 + 1 = 3, then unicorns exist.

b) If 1 + 1 = 3, then dogs can fly.

c) If 1 + 1 = 2, then dogs can fly.

d) If 2 + 2 = 4, then 1 + 2 = 3.


To determine whether these conditional statements are true or false, we need to understand that a conditional statement "If \( P \), then \( Q \)" (denoted as \( P \rightarrow Q \)) is true in all cases except when \( P \) is true and \( Q \) is false. In logic, a conditional statement with a false antecedent \( P \) is considered true regardless of the truth value of the consequent \( Q \).

Let's analyze each statement:

### a) If \( 1 + 1 = 3 \), then unicorns exist.
- \( 1 + 1 = 3 \) is **false**.
- The statement "unicorns exist" is **false** (assuming we're speaking literally).
- Since the antecedent is false, the conditional statement is **true**.

### b) If \( 1 + 1 = 3 \), then dogs can fly.
- \( 1 + 1 = 3 \) is **false**.
- The statement "dogs can fly" is **false** (assuming we're speaking literally).
- Since the antecedent is false, the conditional statement is **true**.

### c) If \( 1 + 1 = 2 \), then dogs can fly.
- \( 1 + 1 = 2 \) is **true**.
- The statement "dogs can fly" is **false**.
- Since the antecedent is true and the consequent is false, the conditional statement is **false**.

### d) If \( 2 + 2 = 4 \), then \( 1 + 2 = 3 \).
- \( 2 + 2 = 4 \) is **true**.
- \( 1 + 2 = 3 \) is **true**.
- Since both the antecedent and consequent are true, the conditional statement is **true**.

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

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is the truth table for a conditional statement?
2. How does the truth value of the antecedent affect a conditional statement?
3. Can a conditional statement be true if the consequent is false? Why or why not?
4. What is the difference between a conditional statement and a biconditional statement?
5. How are conditional statements used in mathematical proofs?

**Tip:** Remember that a conditional statement with a false antecedent is always true, regardless of the truth value of the consequent. This is a fundamental concept in logic.

Explore the truth values of conditional statements with examples like 'If 1 + 1 = 3, then unicorns exist'. Understand the logic behind conditional statements and their truth evaluations. Suitable for Grades 9-12.

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