Question

Which of the following are true? Assume that \(x\) and \(y\) are real numbers. (a) For every \(x, x>0 \Rightarrow x^{2}>0\). (b) For every \(x, x>0 \Leftrightarrow x^{2}>0\). (c) For every \(x, x^{2}>x\). (d) For every \(x\), there exists a \(y\) such that \(y>x^{2}\). (e) For every positive number \(y\), there exists another positive number \(x\) such that \(0

Short answer

Statements (a), (b), (d), and (e) are true; statement (c) is false.

Step-by-step solution

Analyze Statement (a)

Statement (a) says that if \(x > 0\), then \(x^2 > 0\). For any positive real number \(x\), its square, \(x^2 = x \cdot x\), is also positive. Thus, statement (a) is true.

Analyze Statement (b)

Statement (b) is a bi-conditional: \(x > 0 \Leftrightarrow x^2 > 0\). As seen in Step 1, if \(x > 0\), then \(x^2 > 0\). Also, \(x^2 > 0\) implies \(x > 0\) for real numbers \(x\). Therefore, statement (b) is true.

Analyze Statement (c)

Statement (c) claims that for every \(x\), \(x^2 > x\). If \(x = 0\), then \(0^2 = 0\) which is not greater than \(0\). Also, for \(0 < x < 1\), \(x^2 < x\). Therefore, statement (c) is false.

Analyze Statement (d)

Statement (d) asserts that for every \(x\), there exists a \(y\) such that \(y > x^2\). For any real number \(x\), you can choose \(y = x^2 + 1\), satisfying \(y > x^2\). Therefore, statement (d) is true.

Analyze Statement (e)

Statement (e) suggests that for every positive number \(y\), there exists another positive number \(x\) such that \(0 < x < y\). Given a positive \(y\), you can choose \(x = \frac{y}{2}\), which satisfies \(0 < x < y\). Therefore, statement (e) is true.

Key concepts

Conditional Statements in Mathematics

Conditional statements are logical constructs used frequently in mathematics. They take the form "if...then..." and express a condition and a result. In mathematical notation, this is written as \( p \Rightarrow q \), meaning "if \( p \), then \( q \)." Here, \( p \) is the hypothesis, and \( q \) is the conclusion.

It's important to understand conditional statements because they form the backbone of logical arguments in math. For instance, consider the statement "if \( x > 0 \), then \( x^2 > 0 \)" from our example. This tells us that whenever \( x \) is a positive real number, its square is also positive.

To analyze conditional statements, we often consider their truth values under different circumstances. In our example, for any positive \( x \), squaring it results in a positive number. Thus, the conditional statement is true across its domain.

When solving mathematical problems, ensure you thoroughly understand the conditions specified by such statements. This practice will help you apply them accurately and logically.

Bi-Conditional Statements

Bi-conditional statements extend the concept of conditional statements. They work in both directions: "if and only if." This is expressed as \( p \Leftrightarrow q \), meaning "\( p \) if and only if \( q \)." Here, both \( p \Rightarrow q \) and \( q \Rightarrow p \) must be true for the statement to hold.

In the given problem, the statement "\( x > 0 \Leftrightarrow x^2 > 0 \)" is a bi-conditional. It requires us to check two conditions:

• \( x > 0 \Rightarrow x^2 > 0 \)
• \( x^2 > 0 \Rightarrow x > 0 \)

Here's why both directions are true: we already know from the conditional statement analysis that if \( x > 0 \), then \( x^2 > 0 \). Also, when \( x^2 > 0 \), \( x \) cannot be zero or negative, as these would result in \( x^2 \) being zero or potentially positive \( x^2 \). Therefore, \( x \) must be positive to satisfy \( x^2 > 0 \).

Bi-conditionals can be tricky but are crucial for defining equivalences in math. They ensure both sides of the statement are completely aligned.

Inequalities

Inequalities express a relationship of order between two quantities. In the context of the exercise, they play a crucial role, especially in statement (c) where we are asked to determine if "\( x^2 > x \)" for every \( x \).

Understanding inequalities is about comparing quantities. If \( a > b \), \( a \) is greater than \( b \). To validate inequalities, it's necessary to test the relationship across suitable values. In the given statement, testing various ranges like \( x = 0 \) or \( 0 < x < 1 \) shows that \( x^2 \) is not always greater than \( x \). Thus, the statement doesn't hold for all real numbers.

Grasp the nature of inequalities by checking different cases, especially boundary and edge cases. This approach provides insight into where an inequality holds or fails. Familiarity with solving and interpreting inequalities is foundational for solving algebraic expressions and real-world problems.

Existence of Numbers

In mathematics, asserting the existence of numbers that satisfy certain conditions is a common practice, particularly within existential quantifiers. Statement (d) explores this idea by claiming that for every real number \( x \), there exists a \( y \) such that \( y > x^2 \).

This type of statement emphasizes finding at least one value of \( y \) for each \( x \) that meets the criterion. For example, choosing \( y = x^2 + 1 \) works, as it will always be greater than \( x^2 \). This shows that the statement has a true solution via construction.

Similarly, statement (e) discusses the existence of a positive \( x \) smaller than a given positive \( y \). Using \( x = \frac{y}{2} \) guarantees \( 0 < x < y \), fulfilling the statement's requirement.

Understanding these existential proofs involves constructing examples or leveraging known properties. They assure you of solutions and guide you in proving more complex mathematical theorems.