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In the definition of a group on mathworld, http://mathworld.wolfram.com/Group.html" , implies closure, so, isn't it unnecessary to talk about the property closure in the definition of a group?
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AKG said:What do you mean "isn't it sufficient to talk about the property of closure in the definition of a group?" It is not necessary to talk about the property of closure in the defintion of a group. But "not necessary" is not the same thing as "sufficient". Anyways, although it is not necessary, in theory, to talk about the property of closure, you are often just given a set S with a function * with domain SxS, and you have to verify that * is indeed an operation, that is, that closure does indeed hold.
"Group Definition: Closure Not Required" is a concept in mathematics that describes a set of elements that do not necessarily need to produce a new element when combined. This means that the set is closed under the operation defined on it, and therefore does not require closure to be considered a group.
In a regular group, closure is required, meaning that combining any two elements in the group will always result in another element within the group. However, in a group with closure not required, there may be combinations of elements that do not produce a new element, but the set is still considered a group because it is closed under the defined operation.
Yes, it is possible for a group to have closure not required for one operation, but not for another. For example, a set of integers under addition has closure not required, since adding two integers may not always result in another integer. However, the same set under multiplication does require closure, as multiplying two integers will always produce another integer.
One example is the set of all rational numbers (fractions) under addition. While adding two rational numbers may not always result in another rational number, the set is still closed under addition and thus considered a group.
Group theory, including the concept of closure not required, has various applications in fields such as physics, chemistry, and computer science. For example, in quantum mechanics, symmetries of a system can be described using group theory, including groups with closure not required. In chemistry, group theory is used to classify and predict properties of molecules. In computer science, group theory is used in cryptography to develop secure algorithms.