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iv Vrvy(star(r)^ ¬shade(r) A next_to(x, y) → shade(y) A circ(y)) v Vy3r next to(r, y) A shade(r) vi 3rVy next_to(x, y) A shade(r)

iv Vrvy(star(r)^ ¬shade(r) A next_to(x, y) → shade(y) A circ(y)) v Vy3r next to(r, y) A shade(r) vi 3rVy next_to(x, y) A shade(r)

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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**Transcription of Logical Expressions** The image contains logical expressions with quantifiers and predicates, likely representing conditions or statements in formal logic or a programming context. 1. **Expression iv**: \[ \forall x \forall y ( \text{star}(x) \land \neg \text{shade}(x) \land \text{next\_to}(x, y) \rightarrow \text{shade}(y) \land \text{circ}(y) ) \] - This expression states that for all elements \( x \) and \( y \), if \( x \) is a star, not shaded, and is next to \( y \), then \( y \) is shaded and circular. 2. **Expression v**: \[ \forall y \exists x ( \text{next\_to}(x, y) \land \text{shade}(x) ) \] - This expression indicates that for every element \( y \), there exists an element \( x \), such that \( x \) is next to \( y \) and is shaded. 3. **Expression vi**: \[ \exists x \forall y ( \text{next\_to}(x, y) \land \text{shade}(x) ) \] - This expression implies there exists an element \( x \) such that for all elements \( y \), \( x \) is next to \( y \) and is shaded. These expressions use logical quantifiers: - **\(\forall\)** (for all) signifies that the statement holds true for all possible elements. - **\(\exists\)** (there exists) signifies that there is at least one element for which the statement is true. - **\(\land\)** represents the logical AND. - **\(\neg\)** represents the logical NOT. - **\(\rightarrow\)** represents implication, meaning if the condition before it is true, then the condition after it must also be true.
Transcribed Image Text:**Transcription of Logical Expressions** The image contains logical expressions with quantifiers and predicates, likely representing conditions or statements in formal logic or a programming context. 1. **Expression iv**: \[ \forall x \forall y ( \text{star}(x) \land \neg \text{shade}(x) \land \text{next\_to}(x, y) \rightarrow \text{shade}(y) \land \text{circ}(y) ) \] - This expression states that for all elements \( x \) and \( y \), if \( x \) is a star, not shaded, and is next to \( y \), then \( y \) is shaded and circular. 2. **Expression v**: \[ \forall y \exists x ( \text{next\_to}(x, y) \land \text{shade}(x) ) \] - This expression indicates that for every element \( y \), there exists an element \( x \), such that \( x \) is next to \( y \) and is shaded. 3. **Expression vi**: \[ \exists x \forall y ( \text{next\_to}(x, y) \land \text{shade}(x) ) \] - This expression implies there exists an element \( x \) such that for all elements \( y \), \( x \) is next to \( y \) and is shaded. These expressions use logical quantifiers: - **\(\forall\)** (for all) signifies that the statement holds true for all possible elements. - **\(\exists\)** (there exists) signifies that there is at least one element for which the statement is true. - **\(\land\)** represents the logical AND. - **\(\neg\)** represents the logical NOT. - **\(\rightarrow\)** represents implication, meaning if the condition before it is true, then the condition after it must also be true.
**Educational Website Content: Logical Reasoning and Predicate Understanding** The image features a grid of 3x3 squares, each containing a symbol: either a circle, star, or square. Some symbols are shaded, and they are labeled with letters from 'a' to 'g'. The setup is used to form logical statements based on the properties of the symbols and their positions. **Grid Explanation:** - Row 1: - Cell 1 contains a circle labeled 'a'. - Cell 2 contains a star labeled 'b'. - Cell 3 is empty. - Row 2: - Cell 1 contains a star labeled 'c'. - Cell 2 is empty. - Cell 3 contains a circle labeled 'd' (shaded). - Row 3: - Cell 1 contains a circle labeled 'e'. - Cell 2 contains a square labeled 'f'. - Cell 3 contains a square labeled 'g' (shaded). **Predicates:** 1. `square(x)`: True if x is a square; false otherwise. 2. `star(x)`: True if x is a star; false otherwise. 3. `circ(x)`: True if x is a circle; false otherwise. 4. `shade(x)`: True if x is shaded; false otherwise. 5. `next_to(x, y)`: True if x and y are adjacent horizontally, vertically, or diagonally (not reflexive for any object). **Task:** Determine whether the following statements are true or false based on the predicates and grid setup. Justifications are not required for responses: - Analyze each statement using the grid and provided predicate definitions. - Consider adjacency and properties of each labeled symbol. This exercise enhances skills in logical reasoning, understanding predicates, and spatial analysis in a grid context.
Transcribed Image Text:**Educational Website Content: Logical Reasoning and Predicate Understanding** The image features a grid of 3x3 squares, each containing a symbol: either a circle, star, or square. Some symbols are shaded, and they are labeled with letters from 'a' to 'g'. The setup is used to form logical statements based on the properties of the symbols and their positions. **Grid Explanation:** - Row 1: - Cell 1 contains a circle labeled 'a'. - Cell 2 contains a star labeled 'b'. - Cell 3 is empty. - Row 2: - Cell 1 contains a star labeled 'c'. - Cell 2 is empty. - Cell 3 contains a circle labeled 'd' (shaded). - Row 3: - Cell 1 contains a circle labeled 'e'. - Cell 2 contains a square labeled 'f'. - Cell 3 contains a square labeled 'g' (shaded). **Predicates:** 1. `square(x)`: True if x is a square; false otherwise. 2. `star(x)`: True if x is a star; false otherwise. 3. `circ(x)`: True if x is a circle; false otherwise. 4. `shade(x)`: True if x is shaded; false otherwise. 5. `next_to(x, y)`: True if x and y are adjacent horizontally, vertically, or diagonally (not reflexive for any object). **Task:** Determine whether the following statements are true or false based on the predicates and grid setup. Justifications are not required for responses: - Analyze each statement using the grid and provided predicate definitions. - Consider adjacency and properties of each labeled symbol. This exercise enhances skills in logical reasoning, understanding predicates, and spatial analysis in a grid context.
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