Questions: Consider this conditional statement. If Nicole was born on Christmas Day, then Nicole was born on December 25. (a) Give the contrapositive of the statement. (b) Give the converse of the statement. (c) Give the inverse of the statement.

Solution Steps

To solve this problem, we need to understand the logical transformations of a conditional statement. The original statement is "If Nicole was born on Christmas Day, then Nicole was born on December 25."

(a) The contrapositive of a statement "If P, then Q" is "If not Q, then not P."

(b) The converse of a statement "If P, then Q" is "If Q, then P."

(c) The inverse of a statement "If P, then Q" is "If not P, then not Q."

The original conditional statement is: \[ P: \text{Nicole was born on Christmas Day} \] \[ Q: \text{Nicole was born on December 25} \] This can be expressed as: \[ P \implies Q \]

The contrapositive of the statement \( P \implies Q \) is given by: \[ \neg Q \implies \neg P \] This translates to: \[ \text{If not } Q, \text{ then not } P \implies \text{If not Nicole was born on December 25, then not Nicole was born on Christmas Day} \]

The converse of the statement \( P \implies Q \) is: \[ Q \implies P \] This translates to: \[ \text{If } Q, \text{ then } P \implies \text{If Nicole was born on December 25, then Nicole was born on Christmas Day} \]

The inverse of the statement \( P \implies Q \) is: \[ \neg P \implies \neg Q \] This translates to: \[ \text{If not } P, \text{ then not } Q \implies \text{If not Nicole was born on Christmas Day, then not Nicole was born on December 25} \]

Final Answer