Beginning Activity 2: Variables

Beginning Activity Beginning Activity 2: Variables
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Not all mathematical sentences are statements. For example, an equation such as

x^{2} - 5 = 0

is not a statement. In this sentence, the symbol $x$ is a variable. It represents a number that may be chosen from some specified set of numbers. The sentence (equation) becomes true or false when a specific number is substituted for $x .$

1.

Does the equation $x^{2} - 25 = 0$ become a true statement...

(a)

if $- 5$ is substituted for $x ?$

(b)

if $\sqrt{5}$ is substituted for $x ?$

Definition.

A variable is a symbol representing an unspecified object that can be chosen from a given set $U .$ The set $U$ is called the universal set for the variable. It is the set of specified objects from which objects may be chosen to substitute for the variable. A constant is a specific member of the universal set.

Some sets that we will use frequently are the usual number systems. Recall that we use the symbol $R$ to stand for the set of all real numbers, the symbol $Q$ to stand for the set of all rational numbers, the symbol $Z$ to stand for the set of all integers, and the symbol $N$ to stand for the set of all natural numbers.

2.

What real numbers will make the sentence “ $y^{2} - 2 y - 15 = 0$ ” a true statement when substituted for $y ?$

3.

What natural numbers will make the sentence “ $y^{2} - 2 y - 15 = 0$ ” a true statement when substituted for $y ?$

4.

What real numbers will make the sentence “ $\sqrt{x}$ is a real number” a true statement when substituted for $x ?$

5.

What real numbers will make the sentence “ $\sin^{2} x + \cos^{2} x = 1$ ” a true statement when substituted for $x ?$

6.

What natural numbers will make the sentence “ $\sqrt{n}$ is a natural number” a true statement when substituted for $n ?$

7.

What real numbers will make the sentence

\int_{0}^{y} t^{2} d t > 9

a true statement when substituted for $y ?$

Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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Beginning Activity Beginning Activity 2: Variables
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Beginning Activity Beginning Activity 2: Variables A4US

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Beginning Activity Beginning Activity 2: Variables
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