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Which of the following are true in the given universe? Explain your answer. (a) (Vx € N) (x + x ≥x) True or False Why? (b) (VaR) (x² + 6x +520) True or False Why? (c) (3x € N) (2x + 3 = 6x + 7) True or False Why?

Which of the following are true in the given universe? Explain your answer. (a) (Vx € N) (x + x ≥x) True or False Why? (b) (VaR) (x² + 6x +520) True or False Why? (c) (3x € N) (2x + 3 = 6x + 7) True or False Why?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter1: Line And Angle Relationships
Section1.5: The Format Proof Of A Theorem
Problem 11E: When can a theorem be cited as a reason reason in a proof?
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**Logical Statements and Their Verification** This section involves evaluating the truth values of statements under given conditions. Let's explore which of the following are true in the given universe, and provide explanations for each. (a) \((\forall x \in \mathbb{N})(x + x \geq x)\) True or False **Why?** To determine if this statement is true, consider the nature of natural numbers (\(\mathbb{N}\)). For all \(x \in \mathbb{N}\), we need to check if \(x + x \geq x\). (b) \((\forall x \in \mathbb{R})(x^2 + 6x + 5 \geq 0)\) True or False **Why?** This statement requires us to verify whether for all \(x \in \mathbb{R}\), the expression \(x^2 + 6x + 5\) is always greater than or equal to zero. This can be done by examining the quadratic equation and determining its roots. (c) \((\exists x \in \mathbb{N})(2x + 3 = 6x + 7)\) True or False **Why?** We need to determine if there exists an \(x \in \mathbb{N}\) that satisfies the equation \(2x + 3 = 6x + 7\). Solving the equation for \(x\) and checking if the solution is a natural number will provide the answer. For each part, you are required to solve or reason through the expressions and confirm their validity within the defined set (natural numbers for (a) and (c), real numbers for (b)).
Transcribed Image Text:**Logical Statements and Their Verification** This section involves evaluating the truth values of statements under given conditions. Let's explore which of the following are true in the given universe, and provide explanations for each. (a) \((\forall x \in \mathbb{N})(x + x \geq x)\) True or False **Why?** To determine if this statement is true, consider the nature of natural numbers (\(\mathbb{N}\)). For all \(x \in \mathbb{N}\), we need to check if \(x + x \geq x\). (b) \((\forall x \in \mathbb{R})(x^2 + 6x + 5 \geq 0)\) True or False **Why?** This statement requires us to verify whether for all \(x \in \mathbb{R}\), the expression \(x^2 + 6x + 5\) is always greater than or equal to zero. This can be done by examining the quadratic equation and determining its roots. (c) \((\exists x \in \mathbb{N})(2x + 3 = 6x + 7)\) True or False **Why?** We need to determine if there exists an \(x \in \mathbb{N}\) that satisfies the equation \(2x + 3 = 6x + 7\). Solving the equation for \(x\) and checking if the solution is a natural number will provide the answer. For each part, you are required to solve or reason through the expressions and confirm their validity within the defined set (natural numbers for (a) and (c), real numbers for (b)).
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