Consider the set $(-7,4]$ (in interval notation). The set in set-builder notation is $\square$.
Answer (1)
The problem provides a set in interval notation: ( − 7 , 4 ] .
Convert the interval notation to set-builder notation.
The set-builder notation is { x ∣ − 7 < x ≤ 4 } .
The final answer is { x ∣ − 7 < x ≤ 4 } .
Explanation
Understanding Interval Notation The given set is represented in interval notation as ( − 7 , 4 ] . This notation signifies all real numbers that are greater than − 7 and less than or equal to 4 . In set-builder notation, we express this set by defining the conditions that its elements must satisfy.
Converting to Set-Builder Notation To convert the interval notation ( − 7 , 4 ] to set-builder notation, we use the following format: { x ∣ condition on x } . In this case, the condition is that x must be greater than − 7 and less than or equal to 4 . Therefore, we write this as: { x ∣ − 7 < x ≤ 4 }
Final Answer The set in set-builder notation is { x ∣ − 7 < x ≤ 4 } .
Examples
Set-builder notation is useful in many areas of mathematics, including calculus and analysis. For instance, when defining the domain of a function, we often use set-builder notation to specify the set of all possible input values. For example, if we have a function f ( x ) = x − 2 , we can define its domain as { x ∣ x ≥ 2 } , which means the function is only defined for x values greater than or equal to 2. This notation helps to clearly and concisely define the valid inputs for the function.
