Is the Set with Given Conditions an Empty Set?

In this article, we will explore the concept of set theory, particularly focusing on whether a given set is empty. We will analyze the set defined by the conditions ({x : x^2 9 text{ and } 2x 4}) and determine if it is an empty set. We will also consider different number systems and their implications.

Solving the Equations

To determine whether the set ({x : x^2 9 text{ and } 2x 4}) is empty, we need to solve each equation separately and then find the intersection of the solutions.

Solve (x^2 9):

From the equation (x^2 9), we have:

(x 3) or (x -3)

Solve (2x 4):

From the equation (2x 4), we have:

(x 2)

Finding the Intersection:

Next, we need to find the intersection of the solutions from the two equations. The solutions from the first equation are (x 3) and (x -3). The solution from the second equation is (x 2).

Clearly, the solutions from the first equation ((3, -3)) do not include the solution from the second equation (2). Since there are no common solutions between the two equations, the set ({x : x^2 9 text{ and } 2x 4}) is indeed an empty set.

Thus, the statement is true: the set is empty.

Considerations in Different Number Systems

It is important to note that the answer may vary depending on the number system in which we consider the solutions.

Integers (mathbb{Z}): If (x in mathbb{Z}), the set ({x : x^2 9 text{ and } 2x 4}) is indeed empty because there are no integers that satisfy both conditions simultaneously.

Integers Modulo 2 (mathbb{Z}_2): If we consider the set in (mathbb{Z}_2), we can see that:

(x^2 1) because (1 equiv 1 pmod{2}) (2x 2) because (2 equiv 0 pmod{2})

Thus, (x overline{1}) in (mathbb{Z}_2), which means the equivalence class of 1 in (mathbb{Z}_2). Since 1 is congruent to 9 modulo 2 and 4 is congruent to 2 modulo 2, the set is not empty in this context.

Other Groups: Considerations in other groups, such as Z/1Z, can make the set non-empty. Z/1Z is quite trivial and does not provide any additional solutions, but other groups can indeed make the set non-empty. For example, in Z/4Z, the element 1 satisfies both conditions.

Conclusion

The set ({x : x^2 9 text{ and } 2x 4}) is empty in the standard number systems like (mathbb{Z}), but it can be non-empty in other number systems such as (mathbb{Z}_2). It is important to specify the number system to provide a definitive answer.

In ZF set theory, the empty set is unique, so it is better to state that the set is the empty set rather than just that it is empty.

By considering these different perspectives, we can appreciate the nuanced nature of set theory and the importance of context in mathematical statements.