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

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

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.

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 $\angle D F E$.

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 inverse, converse, and contrapositive of the following statement.

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

Write the negation for the statement given below.

Someone trusts everyone.

Construct the truth table and give the corresponding definition for the use of the exclusive or to combine two statements p and q.

p: Sue's house has a white picket fence.

q : Sue's house is brown.

Use the statements above to write a statement whose logical expression is given.

a. not p

b. p and (not q)

Is the following conditional true or false? Explain your reasoning. $\forall$ real numbers $x$, if $x^2=4$, then $x=2$.

A statement is given.

a. Write the negation of the statement.

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

$p$ : For all real numbers $x, \sin x=\sin (-x)$.

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

$$\left\{\begin{array}{l}x^3>8 \\ x \leq 2\end{array} \Rightarrow x>2\right.$$

$$\therefore x^3 \leq 8$$

Let x be a real number. Let p(x), q(x), and r(x) be the following sentences:

$$p(x): x>10 . \quad q(x): x>15 . \quad r(x): x \leq 12 .$$

Write a statement for the following. Simplify the statements, if possible.

(not $r(x))$ and $(\operatorname{not} q(x))$.