6. Determine the truth value of each statement
(a) Artly (x2

Question

6. Determine the truth value of each statement
(a) Artly (x2 <y+ 1) domain: R x R
(b) Axy (x y) domain: Z+ x Z+
(c) Exty (x y) domain: Z+xz+
(d) 3x (x2 <x) domain: Z+
(e) 3x (x2 <x) domain: R

Accepted Answer

Sure, let's determine the truth value of each statement given the specified domains. ### Question 6(a) $$\forall x \forall y (x^2 < y + 1) ext{ domain: } \mathbb{R} imes \mathbb{R}$$ - **Answer Choice:** ***False*** - ∻Key Concept∻ ⚹Universal Quantifier⚹ - ∻Explanation∻ ⚹For all $x$ and $y$ in $\mathbb{R}$, $x^2 < y + 1$ is not true because you can always find an $x$ such that $x^2$ is greater than any given $y + 1$.⚹ ### Question 6(b) $$\forall x \forall y (x \leq y) ext{ domain: } \mathbb{Z}^+ imes \mathbb{Z}^+$$ - **Answer Choice:** ***False*** - ∻Key Concept∻ ⚹Universal Quantifier⚹ - ∻Explanation∻ ⚹For all $x$ and $y$ in $\mathbb{Z}^+$, $x \leq y$ is not true because you can always find $x$ and $y$ such that $x > y$.⚹ ### Question 6(c) $$\exists x \forall y (x \leq y) ext{ domain: } \mathbb{Z}^+ imes \mathbb{Z}^+$$ - **Answer Choice:** ***False*** - ∻Key Concept∻ ⚹Existential and Universal Quantifiers⚹ - ∻Explanation∻ ⚹There does not exist an $x$ in $\mathbb{Z}^+$ such that $x \leq y$ for all $y$ in $\mathbb{Z}^+$. For example, if $x = 1$, there exists $y = 2$ such that $1 \leq 2$ is true, but if $x = 2$, $2 \leq 1$ is false.⚹ ### Question 6(d) $$\exists x (x^2 < x) ext{ domain: } \mathbb{Z}^+$$ - **Answer Choice:** ***False*** - ∻Key Concept∻ ⚹Existential Quantifier⚹ - ∻Explanation∻ ⚹For $x$ in $\mathbb{Z}^+$, $x^2 < x$ is never true because $x^2$ is always greater than or equal to $x$ for all positive integers.⚹ ### Question 6(e) $$\exists x (x^2 < x) ext{ domain: } \mathbb{R}$$ - **Answer Choice:** ***True*** - ∻Key Concept∻ ⚹Existential Quantifier⚹ - ∻Explanation∻ ⚹For $x$ in $\mathbb{R}$, there exist values such as $0 < x < 1$ where $x^2 < x$ is true. For example, if $x = 0.5$, then $0.5^2 = 0.25 < 0.5$.⚹◊What is the role of quantifiers like A, E, and 3 in mathematical logic when evaluating truth values of statements⍭ Generate me a similar question◊

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