In logic, converse, inverse, and contrapositive refer to different forms of conditional statements derived from a given statement. Suppose we have a conditional statement:

• Original Statement (Conditional): "If $p$, then $q$" (symbolically, $p \rightarrow q$).

Now, let’s define each form based on this conditional statement:

1. Converse:

The converse of a statement switches the hypothesis and conclusion. So, for the statement $p \rightarrow q$, the converse is:

• Converse: "If $q$, then $p$" (symbolically, $q \rightarrow p$).

2. Inverse:

The inverse of a statement negates both the hypothesis and the conclusion. So, for the statement $p \rightarrow q$, the inverse is:

• Inverse: "If not $p$, then not $q$" (symbolically, $\neg p \rightarrow \neg q$).

3. Contrapositive:

The contrapositive is a combination of both the converse and the inverse: it negates and switches both the hypothesis and the conclusion. So, for the statement $p \rightarrow q$, the contrapositive is:

• Contrapositive: "If not $q$, then not $p$" (symbolically, $\neg q \rightarrow \neg p$).

Truth Relationships:

• A conditional statement and its contrapositive are logically equivalent (they always have the same truth value).
• Similarly, the converse and the inverse of a statement are logically equivalent to each other, but not necessarily to the original statement.

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Would you like further details on how to apply these concepts or examples of them in use?

Related Questions:

1. What is the difference between logical equivalence and truth tables?
2. Can a converse of a true statement be false?
3. How do you prove the contrapositive of a conditional statement?
4. How do these logic forms apply in mathematical proofs?
5. Can the inverse of a statement ever be equivalent to the original?

Tip:

When trying to disprove a statement, often it's easier to prove its contrapositive rather than the original.