Answer:
Explanation:
A contrapositive is a logical equivalent of the original statement that has the opposite direction and negated conclusion. It can be proven by using the following steps:
Start with the original statement in the form of "If P, then Q".
Negate both the hypothesis (P) and the conclusion (Q) of the original statement.
Conjoin the negated hypothesis and the negated conclusion to form a new statement.
Show that the contrapositive is logically equivalent to the original statement, meaning that it has the same truth value as the original statement for all possible cases.
For example, consider the statement: "If it rains, the roads will be wet." The contrapositive of this statement is "If the roads are not wet, it did not rain." To prove the contrapositive, we can use the following steps:
Start with the original statement "If it rains, the roads will be wet."
Negate both the hypothesis and the conclusion: "It does not rain" and "The roads are not wet".
Conjoin the negated hypothesis and the negated conclusion: "It does not rain and the roads are not wet."
Show that the contrapositive is logically equivalent to the original statement by considering all possible cases:
If it rains, the roads will be wet (as stated in the original statement), and the contrapositive "It does not rain and the roads are not wet" is false.
If it does not rain, the roads will not be wet (as stated in the contrapositive), and the original statement "If it rains, the roads will be wet" is false.
This shows that the contrapositive is logically equivalent to the original statement, and we can use it to draw the same conclusion with a different proof.