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 a negation for each of the following statements. a. Sets A and B do not have any points in common. b. Towns P and Q are not connected by any road on the map.

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.

Write an equation to represent this problem and find the unknown side lengths. Use the 5-D Process to help you organize your thinking and to define your variables, if you need to do so. Remember to define your variable.

(a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Everybody loves somebody.

Give a reason for your answer in each. Assume that all variables represent integers. If n=4k+3, does 8 divide

$$n^2$$

-1?

Give a reason for your answer in each. Assume that all variables represent integers. If n=4k+1, does 8 divide

$$n^2$$

-1?

The numbers are all rational. Write each number as a ratio of two integers.

$$52.4672167216721 \ldots$$

Write a valid argument using the Law of Indirect Reasoning for which the conclusion is false.

Determine whether the statement is true or false. Justify your answer with a proof or a counterexample, as appropriate. In each case use only the definitions of the terms and the Assumptions listed on page 146 not any previously established properties. The negative of any odd integer is odd.

Determine whether the statement is true or false. Justify your answer. A circle is a degenerate conic.

Determine whether the property is true for all integers, true for no integers, or true for some integers and false for other integers. Justify your answers.

$$- a ^ { n } = ( - a ) ^ { n }$$

Use properties of translations to prove each theorem. a. Corresponding Angles Theorem. b. Corresponding Angles Converse.

Rewrite the following statements informally in at least two different ways without using variables or quantifiers. a.

$$\forall$$

rectangles x, x is a quadrilateral. b.

$$\exists$$

a set A such that A has 16 subsets.