Fill in the blanks with the correct numerical values that will complete this set description.
[tex]$\begin{array}{l}
{5,10,15,20,25,30,35,40} \
{p \mid p=}
\end{array}$[/tex]
[ ] A x,
[ ] A [tex]$\leq x \leq$[/tex]
[ ] A) [tex]$x \in W}$[/tex]
Answer (2)
The elements of the set are multiples of 5, so we express each element as p = 5 x .
The smallest element is 5, corresponding to x = 1 , and the largest element is 40, corresponding to x = 8 .
The range of x is 1 ≤ x ≤ 8 , where x is a whole number.
The set can be described as { p ∣ p = 5 x , 1 ≤ x ≤ 8 , x ∈ W } .
Explanation
Understanding the Problem We are given a set { 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 } and we want to describe it using set-builder notation. The set-builder notation will have the form { p ∣ p = □ x , □ ≤ x ≤ □ , x ∈ W } , where W represents the set of whole numbers. Our goal is to find the correct numerical values to fill in the blanks.
Expressing Elements as Multiples of 5 First, we observe that all the elements in the set are multiples of 5. This suggests that we can express each element p as p = 5 x , where x is a whole number.
Determining the Range of x Now, we need to determine the range of values for x . The smallest element in the set is 5, which corresponds to x = 1 since 5 × 1 = 5 . The largest element in the set is 40, which corresponds to x = 8 since 5 × 8 = 40 . Therefore, x ranges from 1 to 8, inclusive.
Final Set Description Putting it all together, we can describe the set as { p ∣ p = 5 x , 1 ≤ x ≤ 8 , x ∈ W } . This means that the first blank should be filled with 5, the second blank should be filled with 1, and the third blank should be filled with 8.
Examples
Understanding sets and set-builder notation is crucial in many areas of mathematics and computer science. For example, in database management, sets are used to define collections of data, and set operations are used to query and manipulate that data. In programming, sets can be used to represent collections of unique elements, such as the set of all users who have access to a particular file. The ability to define sets using set-builder notation allows for concise and precise descriptions of complex data structures.
The set can be described using set-builder notation as {p \mid p = 5x, 1 \leq x \leq 8, x \in W}. Here, the first numerical value is 5, representing the expression for multiples of 5, the second value is 1, indicating the minimum value of x, and the third value is 8, indicating the maximum. This shows that x ranges from 1 to 8 as whole numbers.
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