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i know that when function’s complement is open, then the function is closed and vice versa. but i donnt know to get that. i tried drawing it. but couldnt reach a result. i figured out its not convex.
can someone please help me with this, if possible.
how to approach such types.
thanks :)

shiya , 6 Years ago
Grade 12th pass
anser 1 Answers
Askiitians Tutor Team

Last Activity: 14 Days ago

It sounds like you're delving into some interesting concepts in topology and set theory! The relationship between open and closed sets, particularly in the context of a function's complement, can indeed be a bit tricky at first. Let’s break it down step by step to clarify how this works.

Understanding Open and Closed Sets

In topology, a set is defined as open if, intuitively, you can move a little bit in any direction from any point in the set without leaving the set. Conversely, a set is closed if it contains all its boundary points. A key property of these sets is that the complement of an open set is closed, and the complement of a closed set is open.

Defining Complements

Let’s define what we mean by the complement of a set. If you have a set A within a universal set U, the complement of A, denoted as A', consists of all the elements in U that are not in A. For example:

  • If U = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}.

Open and Closed Sets in Action

Now, let’s consider a function f and its properties. If f is an open function, it means that for every point in the domain, the image of that point under f is an open set. If we take the complement of this function's image, we find that it must be a closed set. This is because the complement of an open set is closed, as mentioned earlier.

Visualizing the Concept

Drawing can indeed help in understanding these concepts. Here’s a way to visualize it:

  • Imagine a number line. Let’s say the open interval (0, 1) is your open set. This means all the points between 0 and 1, but not including 0 and 1 themselves.
  • The complement of this set on the number line would be (-∞, 0] ∪ [1, ∞), which includes the endpoints and is thus a closed set.

Convexity and Its Role

You mentioned that you figured out it’s not convex. Convexity refers to a set where, for any two points within the set, the line segment connecting them lies entirely within the set. Open and closed sets can be convex or non-convex. For instance, the open interval (0, 1) is convex, while the set {0, 1} is closed but not convex. Understanding this distinction can help you visualize and analyze different types of sets.

Approaching Problems Involving Complements

When tackling problems involving open and closed sets, consider the following steps:

  • Identify the set you are working with and determine whether it is open or closed.
  • Calculate the complement of the set within the universal set.
  • Check the properties of the complement to see if it aligns with the definitions of open or closed sets.
  • Use visual aids like diagrams or number lines to help clarify your understanding.

By following these steps, you can systematically approach problems involving the complements of functions and their properties. Keep practicing with different sets, and soon you’ll find that these concepts become much clearer!

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