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.

A statement is given.

a. Write the negation of the statement.

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

$q$: There exists a real number $x$ such that $\sin x+\cos x=1$.

Write the inverse, converse, and contrapositive of the following statement.

If your GPA is below 1.7, then you are ineligible to play sports.

Use a truth table to prove that the inverse error yields an invalid argument.

Write the following statement as an if-then statement. $\forall$ integers $a$ and $b, a-b=\sqrt{3}$ only if $a^2-2 a b+b^2=3$.