Complete the input-output table for the logic network at the right.

$$\begin{array}{ccccc}p & q & p \text { OR } q & \text { NOT } q & \text { (NOT q) AND }(p \text { OR } q) \\ 1 & 1 & ? & ? & ? \\ 1 & 0 & ? & ? & ? \\ 0 & 1 & ? & ? & ? \\ 0 & 0 & ? & ? & ?\end{array}$$

Is this statement true or false? Explain your answer. $\forall$ real numbers $x \exists$ an integer $n$ such that $n x$ is an integer.

Write the negation of the statement. If the statement is written using informal language, then write the negation using informal language.

Every state has at least one member of the House of Representatives.

Is the exclusive or or inclusive or intended in the New Hampshire state motto, Live Free or Die? Explain why you think so.

A statement is given.

a. Write the negation of the statement.

b. Which is true: the given statement or its negation? Justify your answer.

$r$ : No man is an island. (John Donne)

Determine whether it's true or false.

Suppose that $p(n)$ and $q(n)$ are the sentences $p(n)$ : $n$ is a prime number and $q(n): n$ is an odd number.

a. $\forall$ positive integers $n, p(n) \Rightarrow q(n)$.

b. $\forall$ positive integers $n, q(n) \Rightarrow p(n)$.

Write the negation for the statement given below.

$\forall$ nonzero real numbers $x$ and $y, x^2 y^{-2}>0$.

Answer the exercise given below.

$$\begin{aligned} & \left\{\begin{array}{l} \text { Everyone that is at least } 16 \text { years old is eligible } \\ \text { for a driver's license. } \\ \text { Mable is not eligible for a driver's license. } \end{array}\right. \\ & \therefore \text { Mable is not at least 16 years old. } \end{aligned}$$

a. Is this a valid argument? Why or why not?

b. Is this conclusion true? In other words, is it the case that every person who is not eligible for a driver's license is under 16 years of age? Why or why not?

c. Is it possible to answer yes to Part a and no to Part b? Why or why not?

Write the negation of the statement. If the statement is written using informal language, then write the negation using informal language.

Those who cannot remember the past are condemned to repeat it. (George Santayana)

Use the Venn diagram to the right.

a. Write a logical expression for the shaded region.

b. Draw a network where $p, q$, and $r$ are the inputs and the shaded region represents the output.

Rewrite this theorem using two if-then statements:

Two lines are parallel if and only if, when cut by a transversal, corresponding angles have the same measure.

Give an example of two statements $p$ and $q$ such that $q \Rightarrow p$ and $p \Rightarrow q$ have different truth values.

Identify a property that is true for some, but not all, students in your school. Express this as an existential statement.

Draw a Venn diagram representing the statement $p$ exclusive or $q$.