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.