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And, remind ourselves that a mapping connects two sets by establishing pairs. For example the mapping f:A → B, might create the pairs (a,, bą), (a2, b2). (a3. b3), and so on. This pairs each element in A with some element in B. We also know that some mappings between sets are onto, one-to-one or both. You might also note that if we can establish a 1-1 correspondence (onto and one-to-one) between A and B, then we would know the sets were equal. (Can either of explain why?) As we shift to thinking about a group, we have an operation to consider and some special things that have to happen. In other words, in order for the mapping to show the groups are structurally the same (a version of equal), the mapping has to ensure key elements to each other and it has to preserve the operation. What key elements would we want to map to one another? What would have to be true if operation *A in A works in essentially the same way operation *B Works in B?

And, remind ourselves that a mapping connects two sets by establishing pairs. For example the mapping f:A → B, might create the pairs (a,, bą), (a2, b2). (a3. b3), and so on. This pairs each element in A with some element in B. We also know that some mappings between sets are onto, one-to-one or both. You might also note that if we can establish a 1-1 correspondence (onto and one-to-one) between A and B, then we would know the sets were equal. (Can either of explain why?) As we shift to thinking about a group, we have an operation to consider and some special things that have to happen. In other words, in order for the mapping to show the groups are structurally the same (a version of equal), the mapping has to ensure key elements to each other and it has to preserve the operation. What key elements would we want to map to one another? What would have to be true if operation *A in A works in essentially the same way operation *B Works in B?

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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And, remind ourselves that a mapping connects two sets by establishing pairs. For example, the mapping \( f: A \rightarrow B \), might create the pairs \((a_1, b_1), (a_2, b_2), (a_3, b_3)\), and so on. This pairs each element in \( A \) with some element in \( B \). We also know that some mappings between sets are onto, one-to-one or both. You might also note that if we can establish a 1-1 correspondence (onto and one-to-one) between \( A \) and \( B \), then we would know the sets were equal. (Can either of explain why?) As we shift to thinking about a group, we have an operation to consider and some special things that have to happen. In other words, in order for the mapping to show the groups are structurally the same (a version of equal), the mapping has to ensure key elements to each other and it has to preserve the operation. What key elements would we want to map to one another? What would have to be true if operation \(*_A\) in \( A \) works in essentially the same way operation \(*_B\) works in \( B \)?
Transcribed Image Text:And, remind ourselves that a mapping connects two sets by establishing pairs. For example, the mapping \( f: A \rightarrow B \), might create the pairs \((a_1, b_1), (a_2, b_2), (a_3, b_3)\), and so on. This pairs each element in \( A \) with some element in \( B \). We also know that some mappings between sets are onto, one-to-one or both. You might also note that if we can establish a 1-1 correspondence (onto and one-to-one) between \( A \) and \( B \), then we would know the sets were equal. (Can either of explain why?) As we shift to thinking about a group, we have an operation to consider and some special things that have to happen. In other words, in order for the mapping to show the groups are structurally the same (a version of equal), the mapping has to ensure key elements to each other and it has to preserve the operation. What key elements would we want to map to one another? What would have to be true if operation \(*_A\) in \( A \) works in essentially the same way operation \(*_B\) works in \( B \)?
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