Sets A And B: Solving Conditions & Divisors Of 60

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Hey guys! Today, we're diving into a cool math problem involving sets. We'll be figuring out sets A and B based on some specific conditions, and then we'll tackle finding divisors within a set. Buckle up, it's gonna be a fun ride!

Determining Sets A and B Based on Given Conditions

Let's break down the first part of the problem. We need to determine sets A and B that satisfy two conditions simultaneously. Understanding these conditions is key to solving the problem.

Condition A: card(A) = card(A∪B)

This condition tells us something important about the number of elements in set A compared to the union of sets A and B. Let's dissect it. The notation 'card(A)' represents the cardinality of set A, which is just a fancy way of saying the number of elements in set A. Similarly, 'card(A∪B)' represents the cardinality of the union of sets A and B. The union of two sets (A∪B) is a new set that contains all the elements that are in A, or in B, or in both.

So, the condition card(A) = card(A∪B) implies that the number of elements in set A is equal to the number of elements in the combined set A∪B. What does this really mean? Think about it: if adding the elements of B to A doesn't change the total number of elements, it means that all the elements of B must already be present in A. In other words, B must be a subset of A (B ⊆ A). This is a crucial understanding for solving the problem. We've already made a significant leap by interpreting this condition.

Condition B: A = {x ∈ N | 8^2 - 4 < (5^2 - 2^2)x ≤ 10^2 + 5}

Now, let's tackle the second condition, which defines set A using set-builder notation. This might look a bit intimidating at first, but we'll break it down step-by-step. The notation '{x ∈ N | ...}' means "the set of all x belonging to the set of natural numbers (N) such that ..." The part after the vertical bar ('|') is the condition that x must satisfy to be a member of set A.

In our case, the condition is 8^2 - 4 < (5^2 - 2^2)x ≤ 10^2 + 5. This is an inequality involving x. To find the elements of A, we need to solve this inequality for x, keeping in mind that x must be a natural number (positive integer). Let's simplify the inequality:

  • First, calculate the constants: 8^2 - 4 = 64 - 4 = 60, 5^2 - 2^2 = 25 - 4 = 21, and 10^2 + 5 = 100 + 5 = 105.
  • Now the inequality becomes: 60 < 21x ≤ 105.
  • To isolate x, we'll divide all parts of the inequality by 21: 60/21 < x ≤ 105/21.
  • Simplify the fractions: approximately 2.86 < x ≤ 5.

Since x must be a natural number, the possible values for x are 3, 4, and 5. Therefore, set A = {3, 4, 5}. This is a concrete set, and we've determined it based on the given condition. We're making great progress!

Combining the Conditions

Remember that sets A and B must satisfy both conditions simultaneously. We've already figured out that B must be a subset of A (from condition a) and that A = {3, 4, 5} (from condition b). So, what are the possible sets for B? Since B is a subset of A, it can be any combination of elements from A, including the empty set ({}).

The possible subsets of A are:

  • {}
  • {3}
  • {4}
  • {5}
  • {3, 4}
  • {3, 5}
  • {4, 5}
  • {3, 4, 5}

Therefore, any of these sets could be set B, as long as A = {3, 4, 5}. We have successfully determined the sets A and B that meet both conditions! It's all about understanding the conditions, breaking them down, and systematically working towards the solution.

Writing the Set of Divisors of 60 Within a Given Set

Now, let's switch gears and focus on the second part of the problem. We need to write the set formed by the divisors of the number 60 that are found in the set B = {n | n ∈ N and 4 < n < 11}. This is another classic set theory problem that involves understanding divisors and set notation.

Understanding Divisors

First, let's refresh our understanding of divisors. A divisor of a number is an integer that divides the number evenly, leaving no remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without leaving a remainder. Finding the divisors of a number is a fundamental skill in number theory, and it's essential for this problem.

To find the divisors of 60, we can systematically check which numbers divide 60 evenly. Here's the list of all divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. It's always a good idea to be thorough and make sure you haven't missed any divisors.

Defining Set B

Next, we need to understand the set B. The set is defined as B = n | n ∈ N and 4 < n < 11}. This means that B is the set of all natural numbers (n) that are greater than 4 and less than 11. Let's list the elements of set B B = {5, 6, 7, 8, 9, 10. This is a straightforward set of natural numbers within a specific range.

Finding the Common Elements

Now comes the crucial step: we need to find the divisors of 60 that are also elements of set B. In other words, we need to find the intersection of the set of divisors of 60 and set B. This is where our understanding of both divisors and sets comes together.

Let's compare the list of divisors of 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) and the elements of set B (5, 6, 7, 8, 9, 10). The divisors of 60 that are also in set B are 5, 6, and 10. These are the numbers that satisfy both conditions: they divide 60 evenly, and they are within the range defined by set B.

Writing the Final Set

Finally, we can write the set formed by the divisors of 60 that are found in set B. This set is {5, 6, 10}. We have successfully identified the elements that meet both criteria, and we have expressed them as a set. This is the solution to the second part of the problem. It's all about systematically working through the information, understanding the definitions, and applying them step-by-step.

Conclusion

So, there you have it! We've tackled a couple of interesting set theory problems today. We determined sets A and B based on given conditions, and we found the set of divisors of 60 within a specific set. These types of problems are fundamental in mathematics, and they help us develop our logical thinking and problem-solving skills. Remember, the key is to break down the problem into smaller parts, understand the definitions and conditions, and work systematically towards the solution. Keep practicing, and you'll become a set theory pro in no time! You got this, guys!