Conditional and Biconditional Statements

The following is the list of problems for Sections 2.3 and 2.4 of the Book of Proof (pages 44, 46). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

Exercises for Section 2.3

Without changing their meanings, convert each of the following sentences into a sentence having the form “If \( P \), then \( Q \).”

1. A matrix is invertible provided that its determinant is not zero.
2. For a function to be continuous, it is sufficient that it is differentiable.
3. For a function to be integrable, it is necessary that it is continuous.
4. A function is rational if it is a polynomial.
5. An integer is divisible by 8 only if it is divisible by 4.
6. Whenever a surface has only one side, it is non-orientable.
7. A series converges whenever it converges absolutely.
8. A geometric series with ratio \( r \) converges if \( \lvert r \rvert < 1\).
9. A function is integrable provided the function is continuous.
10. The discriminant is negative only if the quadratic equation has no real solutions.
11. You fail only if you stop writing. (Ray Bradbury)
12. People will generally accept facts as truth only if the facts agree with what they already believe. (Andy Rooney)
13. Whenever people agree with me I feel I must be wrong. (Oscar Wilde)

Exercises for Section 2.4

Without changing their meanings, convert each of the following sentences into a sentence having the form “\( P \) if and only if \( Q \).”

1. For a matrix \( A \) to be invertible, it is necessary and sufficient that \( \det(A) \neq 0 \).
2. If a function has a constant derivative then it is linear, and conversely.
3. If \( xy = 0 \) then \( x= 0 \) or \( y=0 \), and conversely.
4. If \( a \in \mathbb{Q} \) then \( 5a \in \mathbb{Q} \), and if \( 5a \in \mathbb{Q} \) then \( a \in \mathbb{Q} \).
5. For an occurrence to become an adventure, it is necessary and sufficient for one to recount it. (Jean-Paul Sartre)