## contrapositive statement

### In mathematics, the contrapositive statement is a logical statement that is formed by switching and negating both the hypothesis and conclusion of an original conditional statement

In mathematics, the contrapositive statement is a logical statement that is formed by switching and negating both the hypothesis and conclusion of an original conditional statement. It is one of the four forms of logical equivalence in propositional logic, along with the original statement, the inverse statement, and the converse statement.

To better understand the concept of a contrapositive statement, let’s start with an example of a conditional statement:

“If it is raining, then the ground is wet.”

The contrapositive statement of the above example would be:

“If the ground is not wet, then it is not raining.”

Notice that the hypothesis (it is raining) and the conclusion (the ground is wet) have been switched and negated in the contrapositive statement.

The contrapositive statement is logically equivalent to the original conditional statement, meaning that if the original statement is true, then the contrapositive statement is also true, and vice versa. This is an important concept in mathematical proofs since it allows us to prove the validity of a statement by proving its contrapositive.

In symbolic notation, the contrapositive statement is often represented by an arrow with a tilde (~) symbol denoting negation. For example, the contrapositive of “p → q” would be “~q → ~p”.

To summarize, the contrapositive statement is a logical statement formed by switching and negating the hypothesis and conclusion of an original conditional statement. It is used in mathematical proofs to establish logical equivalence and can help in proving the validity of a statement.

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Understanding Negation | Exploring the Sign Reversal Operation in MathematicsUnderstanding Converse Statements in Mathematics | Definition, Examples, and Importance

Understanding the Inverse Statement | Explained with Examples and Significance