James Brady said:Homework Statement
Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n##
Homework Equations
What does that last statement mean? Union for n greater than or equal to one times the interval?
The Attempt at a Solution
I can't understand the question.
The "Union of Intervals: A_n" is a mathematical concept that refers to the set of all elements that are contained in any of the intervals in a given sequence of intervals. It is denoted as A_n = [a_1, b_1] ∪ [a_2, b_2] ∪ ... ∪ [a_n, b_n].
The "Union of Intervals: A_n" and the "Intersection of Intervals" are two different mathematical concepts. While the union refers to the set of all elements that are contained in any of the intervals in a given sequence, the intersection refers to the set of all elements that are common to all intervals in the sequence. In other words, the union includes all possible elements, while the intersection includes only those elements that are shared by all intervals.
The "Union of Intervals: A_n" is often used in mathematics and statistics to represent the range of values that a variable can take on. It can also be used to determine the overlapping or non-overlapping areas between two or more sets of data.
To find the "Union of Intervals: A_n", you first need to list out all the intervals in the sequence. Then, you can combine them by including all elements that are present in any of the intervals. This is done by taking the minimum value of the lower bounds of the intervals and the maximum value of the upper bounds of the intervals. The resulting interval is the union of all intervals in the sequence.
Yes, the concept of the "Union of Intervals: A_n" can be applied to infinite sequences of intervals. In such cases, the intervals are usually represented as open intervals (without including the endpoints) or half-open intervals (including one endpoint). The union is then calculated in a similar manner, by taking the minimum and maximum values of the intervals.