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==Groups== | ==Groups== | ||
| − | A group in terms of math is a set of numbers along with a "binary operation" (in this case called a group operation) that has the properties of closure, associativity, identity and inverse. The "binary operation" is any sort of modification to the elements. Examples of these would be the typical operands used in traditional math (+ | + | A group in terms of math is a set of numbers along with a "binary operation" (in this case called a group operation) that has the properties of closure, associativity, identity and inverse. The "binary operation" is any sort of modification to the elements. Examples of these would be the typical operands used in traditional math (+, *). If we were to have group G with elements A, B, C,... and a group operation of addition, then the properties would be applied as follows: |
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Closure: Since A and B are in G and the group operation is "+", the sum of A+B must be contained within the group. | Closure: Since A and B are in G and the group operation is "+", the sum of A+B must be contained within the group. | ||
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| + | Associative: (A + B) + C = A + (B + C) | ||
| + | Identity: A + 0 = A | ||
| + | Inverse: There should be some element that equals <math> A^{-1} </math>, for example, <math> A^{-1} = D </math>. | ||
| − | + | [[ Walther MA271 Fall2020 topic18|Back to Walther MA271 Fall2020 topic18]] | |
[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
Latest revision as of 00:03, 7 December 2020
Groups
A group in terms of math is a set of numbers along with a "binary operation" (in this case called a group operation) that has the properties of closure, associativity, identity and inverse. The "binary operation" is any sort of modification to the elements. Examples of these would be the typical operands used in traditional math (+, *). If we were to have group G with elements A, B, C,... and a group operation of addition, then the properties would be applied as follows:
Closure: Since A and B are in G and the group operation is "+", the sum of A+B must be contained within the group.
Associative: (A + B) + C = A + (B + C)
Identity: A + 0 = A
Inverse: There should be some element that equals $ A^{-1} $, for example, $ A^{-1} = D $.
