Write each of the following sets using set-builder notation:
a. $A=\{11,13,15,17,19\}$
b. $B=\{10,11,12,13,14\}$
c. $C=\{K, i, d\}$

Set A includes odd integers from 11 to 19: { x ∣ x is an odd integer and 11 ≤ x ≤ 19 } .
Set B includes integers from 10 to 14: { x ∣ x ∈ Z , 10 ≤ x ≤ 14 } .
Set C includes the letters K, i, and d: { x ∣ x ∈ { K, i, d }} .
The sets A, B, and C are expressed in set-builder notation as shown above.

Explanation

Understanding the Problem We are given three sets, A, B, and C, and we need to express each of them using set-builder notation. Set-builder notation is a way to define a set by specifying a property that its elements must satisfy.

Expressing Set A For set A = {11, 13, 15, 17, 19}, we observe that the elements are odd numbers between 11 and 19, inclusive. We can express this as: A = { x ∣ x is an odd integer and 11 ≤ x ≤ 19 } Alternatively, we can express the elements as 2 n + 1 for some integer n . When x = 11 , 2 n + 1 = 11 , so n = 5 . When x = 19 , 2 n + 1 = 19 , so n = 9 . Thus, we can also write it as: A = { x ∣ x = 2 n + 1 , n ∈ Z , 5 ≤ n ≤ 9 }

Expressing Set B For set B = {10, 11, 12, 13, 14}, we observe that the elements are consecutive integers from 10 to 14, inclusive. We can express this as: B = { x ∣ x ∈ Z , 10 ≤ x ≤ 14 } Alternatively, we can write: B = { x ∣ x is an integer and 10 ≤ x ≤ 14 }

Expressing Set C For set C = {K, i, d}, we observe that the elements are the letters K, i, and d. We can express this as: C = { x ∣ x ∈ { K, i, d }} Alternatively, we can write: C = { x ∣ x is a letter in the word Kid }

Final Answer Therefore, the sets in set-builder notation are:


a. A = { x ∣ x is an odd integer and 11 ≤ x ≤ 19 } or A = { x ∣ x = 2 n + 1 , n ∈ Z , 5 ≤ n ≤ 9 }
b. B = { x ∣ x ∈ Z , 10 ≤ x ≤ 14 } or B = { x ∣ x is an integer and 10 ≤ x ≤ 14 }
c. C = \{x \mid x \in \{\text{K, i, d}\}\}} or C = { x ∣ x is a letter in the word Kid } $
Examples
Set-builder notation is useful in computer science when defining data structures or specifying conditions for data validation. For example, you might define a set of valid user IDs as all strings that start with a letter and contain only letters and numbers. This can be expressed using set-builder notation, which helps in creating clear and concise rules for data handling and processing.

Answered by GinnyAnswer | 2025-07-08

Set A can be expressed as A = { x ∣ x is an odd integer and 11 ≤ x ≤ 19 } . Set B is represented as B = { x ∣ x ∈ Z , 10 ≤ x ≤ 14 } , and Set C can be written as C = { x ∣ x ∈ { K, i, d }} .
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Answered by Anonymous | 2025-07-16