Or is true. The elements of your sets are sets again. ##\emptyset\, , \,\mathbb{R}\, , \,\{r\,|\,r>a\}## are three sets, but here we consider them as the elements of ##\{\emptyset\, , \,\mathbb{R}\}## and ##\{(a,\infty )\}##. This makes the union a set with three elements, ##\emptyset\, , \,\mathbb{R}\, , \,\{r\,|\,r>a\}##.Ineedhelpimbadatphys said:Homework Statement: the question is about topology, but i just want to know.
isn't {∅,R}∪{]a,∞[:a∈R} equal to {∅,R}
since every member of {]a,∞[:a∈R} is a real number?
or am i just completely misunderstanding unions and intersections?
Relevant Equations: above
above
A group in group theory is a set equipped with a single binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements for each element in the set.
An example of a simple group is the cyclic group of integers modulo n under addition, denoted as \( \mathbb{Z}/n\mathbb{Z} \). For instance, \( \mathbb{Z}/5\mathbb{Z} \) is a simple group where the elements are {0, 1, 2, 3, 4} and the operation is addition modulo 5.
To prove that a set and operation form a group, you need to verify four properties: closure (the operation on any two elements in the set results in another element in the set), associativity (the operation is associative), identity (there exists an element in the set that is an identity element), and inverses (every element in the set has an inverse in the set).
The identity element in a group is an element that, when combined with any element of the group using the group operation, leaves the other element unchanged. For example, in the group of integers under addition, the identity element is 0 because adding 0 to any integer does not change the integer.
The inverse element in a group is significant because it ensures that every element in the group can be "undone" by combining it with its inverse, resulting in the identity element. This property is crucial for the structure of a group, as it guarantees that every element can be paired with another to revert to the identity.