Subsets A B And C Of Universal Set U Prime Numbers And Squares

by Scholario Team 63 views

In this comprehensive exploration, we will delve into the fascinating realm of set theory, focusing on subsets A, B, and C within a defined universal set U. Specifically, we are given a universal set U = x 6 ≤ x ≤ 40, which encompasses all integers from 6 to 40, inclusive. Within this universal set, we have three subsets: A, which comprises all prime numbers within U; B, which includes all numbers whose squares are less than 36; and C, which consists of elements in U that are not present in A. Our primary objective is to meticulously list the elements of each of these subsets, providing a clear and concise understanding of their composition.

This exercise provides a valuable opportunity to reinforce fundamental concepts in set theory, including the definitions of universal sets, subsets, prime numbers, and set complements. By systematically identifying the elements belonging to each subset, we gain a deeper appreciation for the relationships between sets and the principles that govern their interactions. Furthermore, this exploration serves as a practical application of mathematical reasoning and problem-solving skills, which are essential in various fields of study and real-world scenarios.

The universal set U serves as the foundational set encompassing all elements under consideration. In our specific scenario, the universal set U is defined as x 6 ≤ x ≤ 40. This notation signifies that U comprises all integers ranging from 6 to 40, including both 6 and 40. To gain a clearer understanding, let's explicitly list the elements of U:

U = {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}

Understanding the scope of the universal set is crucial as it establishes the boundaries within which we will identify the elements of our subsets A, B, and C. Any element belonging to these subsets must necessarily be a member of the universal set U.

Subset A is defined as the set of all prime numbers within the universal set U. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In simpler terms, a prime number is only divisible by 1 and itself. To identify the elements of A, we need to examine each number in U and determine whether it meets the criteria for primality.

Let's systematically analyze the elements of U:

  • 6 is divisible by 2 and 3, so it is not prime.
  • 7 is only divisible by 1 and 7, so it is prime.
  • 8 is divisible by 2 and 4, so it is not prime.
  • 9 is divisible by 3, so it is not prime.
  • 10 is divisible by 2 and 5, so it is not prime.
  • 11 is only divisible by 1 and 11, so it is prime.
  • 12 is divisible by 2, 3, 4, and 6, so it is not prime.
  • 13 is only divisible by 1 and 13, so it is prime.
  • 14 is divisible by 2 and 7, so it is not prime.
  • 15 is divisible by 3 and 5, so it is not prime.
  • 16 is divisible by 2, 4, and 8, so it is not prime.
  • 17 is only divisible by 1 and 17, so it is prime.
  • 18 is divisible by 2, 3, 6, and 9, so it is not prime.
  • 19 is only divisible by 1 and 19, so it is prime.
  • 20 is divisible by 2, 4, 5, and 10, so it is not prime.
  • 21 is divisible by 3 and 7, so it is not prime.
  • 22 is divisible by 2 and 11, so it is not prime.
  • 23 is only divisible by 1 and 23, so it is prime.
  • 24 is divisible by 2, 3, 4, 6, 8, and 12, so it is not prime.
  • 25 is divisible by 5, so it is not prime.
  • 26 is divisible by 2 and 13, so it is not prime.
  • 27 is divisible by 3 and 9, so it is not prime.
  • 28 is divisible by 2, 4, 7, and 14, so it is not prime.
  • 29 is only divisible by 1 and 29, so it is prime.
  • 30 is divisible by 2, 3, 5, 6, 10, and 15, so it is not prime.
  • 31 is only divisible by 1 and 31, so it is prime.
  • 32 is divisible by 2, 4, 8, and 16, so it is not prime.
  • 33 is divisible by 3 and 11, so it is not prime.
  • 34 is divisible by 2 and 17, so it is not prime.
  • 35 is divisible by 5 and 7, so it is not prime.
  • 36 is divisible by 2, 3, 4, 6, 9, 12, and 18, so it is not prime.
  • 37 is only divisible by 1 and 37, so it is prime.
  • 38 is divisible by 2 and 19, so it is not prime.
  • 39 is divisible by 3 and 13, so it is not prime.
  • 40 is divisible by 2, 4, 5, 8, 10, and 20, so it is not prime.

Therefore, the subset A, consisting of prime numbers within U, is:

A = {7, 11, 13, 17, 19, 23, 29, 31, 37}

Subset B is defined as the set of all numbers x within the universal set U such that _x_² is less than 36. To determine the elements of B, we need to examine each number in U and check if its square is less than 36.

Mathematically, we are looking for values of x in U that satisfy the inequality _x_² < 36. Taking the square root of both sides, we get |x| < 6. However, since our universal set U consists of integers from 6 to 40, there are no elements in U that satisfy this condition. This is because the smallest number in U is 6, and 6² = 36, which is not less than 36.

Therefore, subset B is an empty set, as there are no elements in U whose squares are less than 36.

B = {}

Subset C is defined as the set of all elements in the universal set U that are not present in subset A. In other words, C is the complement of A with respect to U. To identify the elements of C, we need to examine each number in U and determine whether it is also a member of A. If a number is in U but not in A, then it belongs to C.

We have already established the elements of U and A:

U = {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} A = {7, 11, 13, 17, 19, 23, 29, 31, 37}

Now, let's identify the elements of C by removing the elements of A from U:

C = {6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40}

In this detailed exploration, we have successfully identified the elements of subsets A, B, and C within the universal set U = x 6 ≤ x ≤ 40. We determined that subset A, consisting of prime numbers within U, is {7, 11, 13, 17, 19, 23, 29, 31, 37}. Subset B, comprising numbers whose squares are less than 36, is an empty set {}. Finally, subset C, containing elements in U that are not in A, is {6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40}.

This exercise has not only reinforced our understanding of set theory concepts but has also demonstrated the importance of systematic analysis and problem-solving in mathematics. By carefully applying the definitions of prime numbers, set complements, and inequalities, we were able to accurately determine the elements of each subset. This knowledge and skillset are invaluable in various mathematical disciplines and practical applications.