Sets A And B: Find Sets From Union And Intersection

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Hey guys! Today, we're diving into a cool math problem that involves finding sets A and B based on some given conditions. It might seem a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding the Problem

So, the problem gives us four conditions and asks us to figure out exactly what elements are in set A and set B. Let's quickly recap what these conditions mean in set theory:

  • A union B (A ∪ B): This is a new set that includes all the elements that are in A, in B, or in both. Think of it as combining everything from both sets into one big set.
  • A intersection B (A ∩ B): This is a new set that only includes the elements that are in both A and B. It's like finding the overlap between the two sets.

With that refresher in mind, let's jump into the specific conditions we need to work with. We'll take each one and see what clues it gives us about sets A and B. The key here is to go step-by-step and extract as much information as possible from each condition. We will be using a logical approach, making sure that each deduction we make is supported by the given information. Remember, mathematical problem-solving is like detective work – we gather clues and piece them together until we solve the mystery!

Decoding the Conditions

Let's look at the conditions one by one and see what we can figure out.

Condition a: A ∪ B = {x ∈ N / x ≤ 8}

  • A union B gives us the big picture. This condition tells us that when we combine all elements from set A and set B, we get the set of all natural numbers (N) less than or equal to 8. So, A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8}. This is our universe of elements – everything in A and B must come from this set.

This is a crucial piece of information. It defines the scope of our search. We know that neither set A nor set B can contain any elements outside of this set. This is like setting the boundaries of our puzzle – we know all the pieces we need are within this frame.

Condition b: A ∩ B = {3, 5}

  • A intersection B reveals the common ground. This condition tells us that the elements 3 and 5 are present in both set A and set B. This is a direct piece of information, like finding two puzzle pieces that definitely fit together.

These elements, 3 and 5, are the core of the intersection. They are the guaranteed overlaps between the two sets. We can immediately pencil them into our working solution for both A and B. This helps us start building the sets and narrow down the possibilities for the remaining elements.

Condition c: A ∪ {2, 5, 6} = {0, 1, 2, 3, 5, 6, 7}

  • Condition C focuses on set A. This one is interesting. It says that if we combine set A with the set {2, 5, 6}, we get the set {0, 1, 2, 3, 5, 6, 7}. Let's think about what this means. Since 2, 5, and 6 are already in the union, the new elements in the resulting set (0, 1, 3, and 7) must have come from set A.

This condition is a bit more deceptive, but it provides valuable insights into the composition of set A. The elements 0, 1, 3, and 7 are exclusively introduced through set A in this union. This means they must be members of set A, and this is a crucial deduction.

Furthermore, we already know from condition (b) that 3 and 5 are in both A and B. So, combining this information, we are starting to get a clearer picture of what set A looks like.

Condition d: B ∪ {0, 2, 4} = {0, 2, 3, 4, 5, 6, 8}

  • Condition D focuses on set B. Similar to condition (c), this tells us that when we combine set B with {0, 2, 4}, we get {0, 2, 3, 4, 5, 6, 8}. So, the new elements in the resulting set (3, 5, 6, and 8) must have come from set B.

This condition mirrors the logic we applied to condition (c). The elements 3, 5, 6, and 8 are added to the union through set B. Therefore, they must be members of set B. This is a powerful deduction that helps us define set B with increasing accuracy.

Moreover, we already know from condition (b) that 3 and 5 belong to both A and B. So, this reinforces our understanding of the intersection and helps us piece together the full picture of set B.

Putting the Pieces Together

Okay, we've extracted a bunch of information from the conditions. Now, it's time to put it all together and figure out what sets A and B actually are.

  • From condition (c), we know A contains {0, 1, 3, 7}. And from condition (b) we know that 3 and 5 are in both A and B. So, A contains at least {0, 1, 3, 5, 7}.
  • From condition (d), we know B contains {3, 5, 6, 8}. And from condition (b), we know that 3 and 5 are in both A and B. So, B contains at least {3, 5, 6, 8}.

Now, let's revisit condition (a), which tells us that A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8}. We know that A has {0, 1, 3, 5, 7} and B has {3, 5, 6, 8}. Comparing this to the union, we see that the elements 2 and 4 are missing. Let's figure out where they belong.

Looking at condition (c) again, A ∪ {2, 5, 6} = {0, 1, 2, 3, 5, 6, 7}. Since 2 is in the result of the union, but not in the set {2, 5, 6}, it must be in A. So, we add 2 to set A.

Similarly, looking at condition (d), B ∪ {0, 2, 4} = {0, 2, 3, 4, 5, 6, 8}. Since 4 is in the result of the union, but not in the set {0, 2, 4}, it must be in B. So, we add 4 to set B.

The Solution

Alright, drumroll please! Based on all the evidence, we've cracked the case! Here are the sets A and B:

  • A = {0, 1, 2, 3, 5, 7}
  • B = {3, 4, 5, 6, 8}

Checking Our Work

To make sure we're right, let's quickly verify our solution against the original conditions:

  • A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8} (This matches condition a)
  • A ∩ B = {3, 5} (This matches condition b)
  • A ∪ {2, 5, 6} = {0, 1, 2, 3, 5, 6, 7} (This matches condition c)
  • B ∪ {0, 2, 4} = {0, 2, 3, 4, 5, 6, 8} (This matches condition d)

Woohoo! Our solution checks out against all the conditions. We've successfully found sets A and B! Consistent verification is crucial in math, ensuring that the solution aligns with all the given constraints.

Key Takeaways

  • Break it down: Complex problems become manageable when tackled step-by-step.
  • Understand the definitions: Knowing what union and intersection mean is crucial.
  • Extract information: Each condition gives you valuable clues.
  • Logical deduction: Use the clues to build your solution piece by piece.
  • Verification is key: Always check your answer against the original conditions.

So, there you have it! Finding sets from unions and intersections might seem tough, but with a bit of logic and careful deduction, you can totally nail it. Keep practicing, and you'll become a set theory master in no time! Remember, practice makes perfect in mathematics, and each problem you solve strengthens your understanding and skills.