Set Operations & Ordering Numbers: Math Problem Solution

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Hey guys! Today, we're diving deep into a couple of interesting math problems involving set operations and number ordering. We'll break down each problem step-by-step, making sure you understand the logic and techniques involved. So, buckle up and let's get started!

Problem 1: Set Operations (Union, Intersection, Difference)

This problem focuses on set theory, specifically on finding the union, intersection, and difference of two sets. These set operations are fundamental concepts in mathematics and computer science, so grasping them well is super important. We're given two sets, A and B, defined based on the divisors of 12 and 21, respectively. Our mission is to find: A ∪ B (A union B), A ∩ B (A intersection B), A - B (A minus B), and B - A (B minus A).

Defining the Sets

First, let's clearly define sets A and B. Remember, the curly braces {} denote a set, and the elements within represent the members of that set. Set A is defined as all x belonging to the set of integers (C, although typically integers are denoted by Z) such that x is a divisor of 12. So, we need to list all the integers that divide 12 without leaving a remainder. These are: -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, and 12. That's a total of 12 elements in set A.

Set B, on the other hand, is defined as all x belonging to the set of integers (again, C should likely be Z) such that x is a divisor of 21. The divisors of 21 are: -21, -7, -3, -1, 1, 3, 7, and 21. Notice that set B has 8 elements. Now that we have clearly defined both sets, we can move on to performing the set operations.

Calculating A ∪ B (A Union B)

The union of two sets, denoted by A ∪ B, is a new set that contains all the elements that are in A, or in B, or in both. In simpler terms, we combine all the unique elements from both sets into a single set. To find A ∪ B, we'll list all the elements from set A and then add any elements from set B that are not already present in the list.

Set A has: -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12}. Now we look at set B {-21, -7, -3, -1, 1, 3, 7, 21. We see that -21, -7, 7, and 21 are not in A, so we add them to the union. Therefore, A ∪ B = {-21, -12, -7, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 7, 12, 21}. This set is quite large, containing 16 elements in total.

Calculating A ∩ B (A Intersection B)

The intersection of two sets, written as A ∩ B, is the set containing only the elements that are common to both A and B. It's like finding the overlap between the two sets. To find A ∩ B, we need to identify the elements that appear in both the list of divisors of 12 and the list of divisors of 21. Looking back at our sets:

  • Set A: {-12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12}
  • Set B: {-21, -7, -3, -1, 1, 3, 7, 21}

We can see that the common elements are -3, -1, 1, and 3. Therefore, A ∩ B = {-3, -1, 1, 3}. This set is significantly smaller than the union, containing only 4 elements.

Calculating A - B (A Minus B)

The difference between two sets, denoted by A - B, is the set containing all the elements that are in A but not in B. Think of it as removing the elements of B from A. To find A - B, we start with set A and eliminate any elements that are also present in set B.

Starting with set A: -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12}, we remove the elements that are also in B {-21, -7, -3, -1, 1, 3, 7, 21. The common elements -3, -1, 1, and 3 are removed from A. Therefore, A - B = {-12, -6, -4, -2, 2, 4, 6, 12}.

Calculating B - A (B Minus A)

Similarly, B - A represents the set of elements that are in B but not in A. It's the opposite of A - B. To find B - A, we start with set B and remove any elements that are also present in set A.

Starting with set B: -21, -7, -3, -1, 1, 3, 7, 21}, we remove the elements that are also in A {-12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12. The common elements -3, -1, 1, and 3 are removed from B. Therefore, B - A = {-21, -7, 7, 21}.

Summary of Set Operations

To recap, we've successfully calculated the four set operations:

  • A ∪ B = {-21, -12, -7, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 7, 12, 21}
  • A ∩ B = {-3, -1, 1, 3}
  • A - B = {-12, -6, -4, -2, 2, 4, 6, 12}
  • B - A = {-21, -7, 7, 21}

Understanding these set operations is crucial for various areas of mathematics and computer science, including database management, logic, and algorithm design. Good job on mastering these concepts!

Problem 2: Ordering Numbers in Descending Order

Now, let's switch gears and tackle the second problem, which involves ordering numbers in descending order. This problem tests our understanding of absolute values, radicals, and exponents. We're given two numbers, 'a' and 'b', expressed in somewhat complicated forms, and our task is to arrange them from largest to smallest. This might seem daunting at first, but we'll break it down step-by-step to make it manageable. The key here is to simplify each number as much as possible before comparing them. Let's dive into it!

Simplifying Number 'a'

Number 'a' is given by: a = 3|1 - √3| - |3√3 - 5| + |-3||2|. The expression involves absolute values and radicals, so let's tackle each part individually. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. This is a crucial point!

First, consider |1 - √3|. Since √3 is approximately 1.732, 1 - √3 is negative. The absolute value of a negative number is its positive counterpart, so |1 - √3| = √3 - 1. Now we have: a = 3(√3 - 1) - |3√3 - 5| + |-3||2|.

Next, let's look at |3√3 - 5|. Since √3 is approximately 1.732, 3√3 is approximately 5.196, which is greater than 5. Therefore, 3√3 - 5 is positive, and the absolute value doesn't change the sign: |3√3 - 5| = 3√3 - 5. Our expression now becomes: a = 3(√3 - 1) - (3√3 - 5) + |-3||2|.

Finally, let's deal with |-3||2|. The absolute value of -3 is 3, so |-3| = 3. The absolute value of 2 is 2, so |2| = 2. Thus, |-3||2| = 3 * 2 = 6. Our simplified expression is: a = 3(√3 - 1) - (3√3 - 5) + 6.

Now, let's distribute and combine like terms: a = 3√3 - 3 - 3√3 + 5 + 6. Notice that the 3√3 and -3√3 terms cancel each other out. We're left with: a = -3 + 5 + 6 = 8. So, number 'a' simplifies to 8. That's a great simplification! We've managed to reduce a complex expression to a single, clear number.

Simplifying Number 'b'

Number 'b' is given by: b = (2^{(2/3) + 4 * 8 / 3}). This expression involves exponents, so we'll use the rules of exponents to simplify it. The key rule we'll use is that when multiplying numbers with the same base, we add their exponents. Let's tackle the exponent first.

The exponent is (2/3) + 4 * (8/3). We need to follow the order of operations (PEMDAS/BODMAS), so we perform the multiplication before the addition. 4 * (8/3) = 32/3. Now we have: (2/3) + (32/3). Adding these fractions, we get (2 + 32) / 3 = 34/3. So, the exponent simplifies to 34/3. We're making progress!

Now we have: b = 2^(34/3). This is where it gets a little tricky. We can rewrite the exponent as a mixed number: 34/3 = 11 and 1/3. So, b = 2^(11 + 1/3). Using the rule of exponents that says x^(a+b) = x^a * x^b, we can rewrite this as: b = 2^11 * 2^(1/3). 2^11 is 2048. 2^(1/3) is the cube root of 2, which is approximately 1.26.

Therefore, b ≈ 2048 * 1.26, which is approximately 2580.48. So, number 'b' is approximately 2580.48. This is significantly larger than 'a'.

Ordering the Numbers

Now that we've simplified both numbers, ordering them is straightforward. We have:

  • a = 8
  • b ≈ 2580.48

Clearly, b is much larger than a. Therefore, the numbers in descending order are b, a. And we've done it! We've successfully ordered the numbers from largest to smallest.

Conclusion

Wow, we've covered a lot in this article! We tackled set operations, simplifying expressions with absolute values and radicals, and applying the rules of exponents. These problems demonstrate the importance of understanding fundamental mathematical concepts and applying them systematically. By breaking down complex problems into smaller, manageable steps, we can solve even the most challenging questions. Keep practicing, and you'll become a math whiz in no time! Remember, the key is to understand the why behind the what. Keep learning, guys!