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Let X= {a,b,c,d}, Y = {1,2}, and G = {(a, 1), (b, 2), (d, 1)}. Is G the graph 7. of a function from the set X to the set Y? Briefly justify your answer.

Let X= {a,b,c,d}, Y = {1,2}, and G = {(a, 1), (b, 2), (d, 1)}. Is G the graph 7. of a function from the set X to the set Y? Briefly justify your answer.

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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**Question 7** Let \( X = \{a, b, c, d\} \), \( Y = \{1, 2\} \), and \( G = \{(a, 1), (b, 2), (d, 1)\} \). Is \( G \) the graph of a function from the set \( X \) to the set \( Y \)? Briefly justify your answer. **Answer:** To determine if \( G \) is the graph of a function from \( X \) to \( Y \), we must check the definition of a function, which requires that every element in the domain \( X \) is mapped to exactly one element in the codomain \( Y \). - The domain \( X \) has four elements: \( a, b, c, d \). - The graph \( G \) contains pairs: \( (a, 1), (b, 2), (d, 1) \). For \( G \) to be a function from \( X \) to \( Y \): 1. Every element in \( X \) must appear as the first component in exactly one of the pairs in \( G \). **Analysis:** - Element \( a \) maps to 1. - Element \( b \) maps to 2. - Element \( d \) maps to 1. - Element \( c \) does not appear in any pair. Since element \( c \) from \( X \) does not map to any element in \( Y \), \( G \) does not represent a function from \( X \) to \( Y \). A function must have a defined output for every element in the domain. Therefore, \( G \) is not a valid graph of a function from \( X \) to \( Y \).
Transcribed Image Text:**Question 7** Let \( X = \{a, b, c, d\} \), \( Y = \{1, 2\} \), and \( G = \{(a, 1), (b, 2), (d, 1)\} \). Is \( G \) the graph of a function from the set \( X \) to the set \( Y \)? Briefly justify your answer. **Answer:** To determine if \( G \) is the graph of a function from \( X \) to \( Y \), we must check the definition of a function, which requires that every element in the domain \( X \) is mapped to exactly one element in the codomain \( Y \). - The domain \( X \) has four elements: \( a, b, c, d \). - The graph \( G \) contains pairs: \( (a, 1), (b, 2), (d, 1) \). For \( G \) to be a function from \( X \) to \( Y \): 1. Every element in \( X \) must appear as the first component in exactly one of the pairs in \( G \). **Analysis:** - Element \( a \) maps to 1. - Element \( b \) maps to 2. - Element \( d \) maps to 1. - Element \( c \) does not appear in any pair. Since element \( c \) from \( X \) does not map to any element in \( Y \), \( G \) does not represent a function from \( X \) to \( Y \). A function must have a defined output for every element in the domain. Therefore, \( G \) is not a valid graph of a function from \( X \) to \( Y \).
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