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g 1, g 2 denotes the ordered pair of the two group elements. *' can be viewed as the naturally induced addition of +. In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action.
The relationship between opposites is known as opposition. A member of a pair of opposites can generally be determined by the question What is the opposite of X ? The term antonym (and the related antonymy) is commonly taken to be synonymous with opposite, but antonym also has other more restricted meanings. Graded (or gradable) antonyms are ...
The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, − (−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of ...
In category theory, a branch of mathematics, the opposite category or dual category C op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself.
The unity of opposites is the philosophical idea that opposites are interconnected due to the way each is defined in relation to the other. Their interdependence unites the seemingly opposed terms. Their interdependence unites the seemingly opposed terms.
A group is a set with an associative operation that has an identity element, and for which every element has an inverse. Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one. It is also an involution, since the inverse of the inverse of an element is the element itself.
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