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- We usually don't use the notation ∗ ∗ when describing group operations. Instead, we use the multiplication symbol ⋅ ⋅ for the operation in an arbitrary group, and call applying the operation “multiplying”—even though the operation may not be “multiplication” in the non-abstract, traditional sense!
A group (G, *) is a set G together with a binary operation * such that: Closure Law: a*b G for all a, b G. Associative Law: (a * b) * c = a * (b * c) for all a, b, c G. Identity Law:There exists e G such that a*e = a = e*a for all a G. Inverse Law: For all a G there exists b G such that a*b = e = b*a.
When no confusion is possible, notation f(S) is commonly used. 1. Closed interval : if a and b are real numbers such that a ≤ b {\displaystyle a\leq b} , then [ a , b ] {\displaystyle [a,b]} denotes the closed interval defined by them.
the notation $(G,.)$ mean you have a group where the operation is called "." If you write $(G,+)$, the name of the operation is $+$. the $.$ and $+$ are just name for operations.
Make a note that while there exists only one identity for every single element in the group, each element in the group has a different inverse. The notation that we use for inverses is a-1. So in the above example, a-1 = b. In the same way, if we are talking about integers and addition, 5-1 = -5.
English mathematicians use the same set-theoretic language, or notation. Notation. We will use the standard notations of set theory. The nota-tion is important. It is designed so we can say things brie y and precisely. It is part of the language of mathematics. You must learn to use it, and use it correctly.
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Oct 10, 2021 · A group with a single element (which is necessarily the identity element) is called a trivial group. In multiplicative notation, one might write \(\{1\}\text{,}\) and in additive notation, one might write \(\{0\}\text{,}\) to denote a trivial group.
