# MATH 300 Fall 2018 Assignment 08

## 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* \( Q \).”

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

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