Contrapositive
In mathematics, the contrapositive is a logical statement that can be derived from an original implication statement by switching the hypothesis and the conclusion, and negating both of them
In mathematics, the contrapositive is a logical statement that can be derived from an original implication statement by switching the hypothesis and the conclusion, and negating both of them.
To understand the contrapositive, let’s consider an example.
Original Statement: If it is raining, then the streets are wet.
To find the contrapositive, we first switch the hypothesis and the conclusion:
Contrapositive: If the streets are not wet, then it is not raining.
Next, we negate both the hypothesis and the conclusion:
Contrapositive (after negation): If it is not raining, then the streets are not wet.
In the contrapositive, we have essentially reversed the statement and added negations to both the hypothesis and the conclusion.
The contrapositive statement is useful in logic and proof writing because it is logically equivalent to the original statement. This means that if the original statement is true, then the contrapositive statement is also true, and vice versa.
So, in some cases, proving the contrapositive is an easier or more direct route to establish the truth of a statement. If we can prove that the contrapositive is true, then it implies that the original statement must also be true.
In summary, the contrapositive of a statement is obtained by switching the hypothesis and the conclusion, and negating both of them. It is a powerful tool in mathematical reasoning and proof writing.
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