Identify whether the statement given below is true or false.

$\forall$ real numbers $a, b, c, \exists$ exactly two real numbers $x$ such that $a x^2+b x+c=0$. If false, find a counterexample.

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.

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.