Determine whether the argument is valid. Justify your reasoning. When Ty hits three home runs in a game, then his team wins. If Ty was not named most valuable player, then his team did not win. Ty hit three home runs in the game. Therefore, Ty was named most valuable player.

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?

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

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 $n, q(n) \Rightarrow p(n)$.

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)

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

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

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.

Write a conditional statement that is logically equivalent to

In order for $2^x>1$ it is sufficient that $x>0$.

Write the negation for the statement given below.

Someone trusts everyone.

Match the argument with the appropriate argument form and tell whether the argument is valid.

A Law of Detachment B Converse Error C Law of Transitivity D Inverse Error E Law of Indirect F Improper Reasoning Induction

$$\left\{\begin{aligned} \text { If } \triangle A B C & \cong \triangle D E F \text {, then } \\ \angle A C B & \cong \angle D F E .\end{aligned}\right.$$

$\triangle A B C$ is not congruent to $\triangle D E F$.

$\therefore \angle A C B$ is not congruent to $\angle D F E$.

From the previous lesson, there was no truth table for the biconditional $p \Leftrightarrow q$.

a. Complete the following truth table for the biconditional.

b. Complete this statement: In a true biconditional, $p$ and $q$ have (the same, opposite) truth values.