The problem asks to transform the statement:

"If the integer $n$ is even, then $n^2 + 1$ is odd."

This is a conditional statement in logic. Let's break down each part of the solution: disjunctive form, negation, contrapositive, inverse, and converse.

Let’s define:

• $p$: "The integer $n$ is even."
• $q$: "$n^2 + 1$ is odd."

The statement in symbolic form is: $p \rightarrow q$

1. Disjunctive form:

The disjunctive form of an implication $p \rightarrow q$ is logically equivalent to $\neg p \lor q$. This is read as "not $p$ or $q$".

• So, for our case: $\neg p \lor q$ Which translates to: "If $n$ is not even, or $n^2 + 1$ is odd."

2. Negation:

To negate $p \rightarrow q$, we use the fact that the negation of an implication is $p \land \neg q$.

• So, negating the statement: $p \land \neg q$ This translates to: "The integer $n$ is even, and $n^2 + 1$ is not odd."

3. Contrapositive:

The contrapositive of $p \rightarrow q$ is $\neg q \rightarrow \neg p$.

• So, the contrapositive is: "If $n^2 + 1$ is not odd, then $n$ is not even." Which simplifies to: "If $n^2 + 1$ is even, then $n$ is odd."

4. Inverse:

The inverse of $p \rightarrow q$ is $\neg p \rightarrow \neg q$.

• So, the inverse is: "If $n$ is not even, then $n^2 + 1$ is not odd." Which simplifies to: "If $n$ is odd, then $n^2 + 1$ is even."

5. Converse:

The converse of $p \rightarrow q$ is $q \rightarrow p$.

• So, the converse is: "If $n^2 + 1$ is odd, then $n$ is even."

Would you like any further clarification or details on these transformations? Here are 5 related questions that may expand your understanding:

1. What is the difference between the contrapositive and converse of a statement?
2. How do you prove that $\neg p \lor q$ is logically equivalent to $p \rightarrow q$?
3. How do negation rules apply to compound statements like conjunctions and disjunctions?
4. Can the inverse and converse of a statement ever both be true?
5. How can you construct truth tables for implications, converses, and inverses?

Tip: Always remember that the contrapositive of a statement is logically equivalent to the original statement, but the inverse and converse are not guaranteed to be true.