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Match the following symbolic statements with their English equivalents. VVy(x3=y3 → x=y) X(E(x) → E(x+2)) Ey (sin(x)=y) Vy 3x (sin(x)=y) a. For any x there is a y such that sin(x)=y. In other words, every number x is in the domain of sine. b. Any even number plus 2 is an even number. c. For every y there is an x such that sin(x)=y. In other words, every number y is in the range of sine (which is false). d. For any numbers, if the cubes of two numbers are equal, then the numbers are

Match the following symbolic statements with their English equivalents. VVy(x3=y3 → x=y) X(E(x) → E(x+2)) Ey (sin(x)=y) Vy 3x (sin(x)=y) a. For any x there is a y such that sin(x)=y. In other words, every number x is in the domain of sine. b. Any even number plus 2 is an even number. c. For every y there is an x such that sin(x)=y. In other words, every number y is in the range of sine (which is false). d. For any numbers, if the cubes of two numbers are equal, then the numbers are

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Symbolic Statements and Their English Equivalents #### Match the following symbolic statements with their English equivalents: 1. **∀x ∀y (x³=y³ → x=y)** 2. **∀x (E(x) → E(x+2))** 3. **∀x ∃y (sin(x)=y)** 4. **∀y ∃x (sin(x)=y)** #### English Equivalents: a. For any \( x \) there is a \( y \) such that \( \sin(x)=y \). In other words, every number \( x \) is in the domain of sine. b. Any even number plus 2 is an even number. c. For every \( y \) there is an \( x \) such that \( \sin(x)=y \). In other words, every number \( y \) is in the range of sine (which is false). d. For any numbers, if the cubes of two numbers are equal, then the numbers are equal. ### Detailed Explanation: 1. **Statement 1:** - Symbolic: **∀x ∀y (x³=y³ → x=y)** - Translation: This means for any numbers \( x \) and \( y \), if the cube of \( x \) equals the cube of \( y \), then \( x \) is equal to \( y \). 2. **Statement 2:** - Symbolic: **∀x (E(x) → E(x+2))** - Translation: For any number \( x \), if \( x \) is even, then \( x+2 \) is also even. 3. **Statement 3:** - Symbolic: **∀x ∃y (sin(x)=y)** - Translation: For any number \( x \), there exists a number \( y \) such that \( \sin(x) = y \). This reflects that every \( x \) has a sine value. 4. **Statement 4:** - Symbolic: **∀y ∃x (sin(x)=y)** - Translation: For every number \( y \), there exists a number \( x \) such that \( \sin(x) = y \). This suggests that every \( y \) is achievable as a sine value, which is not true since sine values range from -
Transcribed Image Text:### Symbolic Statements and Their English Equivalents #### Match the following symbolic statements with their English equivalents: 1. **∀x ∀y (x³=y³ → x=y)** 2. **∀x (E(x) → E(x+2))** 3. **∀x ∃y (sin(x)=y)** 4. **∀y ∃x (sin(x)=y)** #### English Equivalents: a. For any \( x \) there is a \( y \) such that \( \sin(x)=y \). In other words, every number \( x \) is in the domain of sine. b. Any even number plus 2 is an even number. c. For every \( y \) there is an \( x \) such that \( \sin(x)=y \). In other words, every number \( y \) is in the range of sine (which is false). d. For any numbers, if the cubes of two numbers are equal, then the numbers are equal. ### Detailed Explanation: 1. **Statement 1:** - Symbolic: **∀x ∀y (x³=y³ → x=y)** - Translation: This means for any numbers \( x \) and \( y \), if the cube of \( x \) equals the cube of \( y \), then \( x \) is equal to \( y \). 2. **Statement 2:** - Symbolic: **∀x (E(x) → E(x+2))** - Translation: For any number \( x \), if \( x \) is even, then \( x+2 \) is also even. 3. **Statement 3:** - Symbolic: **∀x ∃y (sin(x)=y)** - Translation: For any number \( x \), there exists a number \( y \) such that \( \sin(x) = y \). This reflects that every \( x \) has a sine value. 4. **Statement 4:** - Symbolic: **∀y ∃x (sin(x)=y)** - Translation: For every number \( y \), there exists a number \( x \) such that \( \sin(x) = y \). This suggests that every \( y \) is achievable as a sine value, which is not true since sine values range from -
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