Question:
The set U ={ x£ R | sinx =1/2}is a. Open b. Closed. c. Both open and closed d. Neither open nor closed. Explain in detail.

Answer:


A set is said to be open if every point in the set is an interior point, which means that there exists an open ball centered at that point which is completely contained in the set.

A set is said to be closed if it contains all of its limit points.


The set U = { x £ R | sinx = 1/2 } is the set of all real numbers x such that the sine of x is equal to 1/2.

To determine whether the set U is open, closed, both open and closed, or neither open nor closed, we need to analyze its properties.



1. Openness:
To determine if the set U is open, we need to check if every point in U is an interior point.

- The equation sinx = 1/2 has infinitely many solutions in the real numbers, such as x = π/6, x = 5π/6, x = π/2, etc.
- For each solution x, there exists an open interval centered at x, such as (x - δ, x + δ), where δ is a positive number, that contains only points whose sine values are also 1/2.
- Therefore, every point in U has an open ball centered at that point which is completely contained in U.
- Thus, the set U is open.

2. Closedness:
To determine if the set U is closed, we need to check if it contains all of its limit points.

- The limit points of the set U are the values of x for which sinx = 1/2. In other words, the values of x where the sine function approaches 1/2.
- The solutions to sinx = 1/2 are x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer.
- By examining these solutions, we can see that every limit point of U is already included in U.
- Therefore, the set U contains all of its limit points and is closed.


Based on the analysis of the set U, we can conclude that it is both open and closed.