Question

if we win the game today, then we will win the tournament.
what is the contrapositive of this statement?
a. if we win the tournament, then we won the game today.
b. if we do not win the game today, then we will not win the tournament.
c. if we win the game today, then we will not win the tournament.
d. if we dont win the tournament, then we didnt win the game today.

Explanation:

To find the contrapositive of a conditional statement \( p \to q \), we negate both the hypothesis and the conclusion and then reverse them, resulting in \(
eg q \to
eg p \).

Let \( p \) be "we win the game today" and \( q \) be "we will win the tournament". The original statement is \( p \to q \).

• The negation of \( q \) (\(

eg q \)) is "we don't win the tournament".

• The negation of \( p \) (\(

eg p \)) is "we didn't win the game today".

Reversing these, the contrapositive is \(
eg q \to
eg p \), which translates to "If we don't win the tournament, then we didn't win the game today".

Now let's analyze the options:

• Option A is the converse (\( q \to p \)), not the contrapositive.
• Option B is the inverse (\(

eg p \to
eg q \)), not the contrapositive.

• Option C is not a valid transformation of the original statement.
• Option D matches \(

eg q \to
eg p \), the contrapositive.

Answer:

D. If we don't win the tournament, then we didn't win the game today.