Set Operations: A∪B, A∩B, A\B, B\A For A=(4,10], B=(5,8)

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Hey guys! Let's dive into some set theory today. We're going to tackle a problem where we need to find the union, intersection, and set differences between two sets, A and B. To make things super clear, we'll also draw a diagram to visualize what's going on. So, grab your thinking caps, and let's get started!

Understanding the Sets

First, let's define our sets. We have set A, which is the interval (4, 10], and set B, which is the interval (5, 8). It's super important to remember what these notations mean:

  • (4, 10] means all numbers greater than 4 and less than or equal to 10. 4 is not included, but 10 is included.
  • (5, 8) means all numbers greater than 5 and less than 8. Neither 5 nor 8 are included.

Visualizing these sets on a number line can be incredibly helpful. Imagine a number line stretching from negative infinity to positive infinity. Set A would be a line segment starting just to the right of 4 (but not including 4) and stretching all the way to 10, where it includes 10. Set B would be a shorter segment inside A, starting just to the right of 5 and ending just before 8.

Now that we understand what our sets look like, let's get into the operations.

A∪B: The Union of A and B

Okay, so what's the union of two sets? Simply put, the union (denoted by ∪) is a new set that contains all the elements from both sets. Think of it like merging two groups of friends – everyone is invited!

In our case, A∪B means we want to combine all the numbers in set A and set B. Since B is partially inside A, we need to consider the outermost boundaries. Set A goes from just above 4 to 10 (inclusive), and set B goes from just above 5 to just below 8. When we combine them, the union will stretch from the leftmost point (just above 4) to the rightmost point (10, inclusive).

Therefore, A∪B = (4, 10]. It's crucial to include the square bracket at 10 because 10 is part of set A. We use a parenthesis at 4 because 4 is not included in set A.

In simple terms, A∪B takes the start of A and the end of A, ignoring B because it falls within those boundaries. Understanding this will make the rest of the operations much easier!

A∩B: The Intersection of A and B

Next up, we have the intersection (denoted by ∩). This operation is a bit more exclusive. The intersection of two sets is a new set containing only the elements that are common to both sets. Think of it as the group of friends who are invited to both parties – only the overlap matters.

For A∩B, we're looking for the numbers that are in both set A and set B. Looking back at our sets: A = (4, 10] and B = (5, 8). Set B is entirely contained within the bounds of set A, but it doesn't start at the same point. The common area starts just after 5 (since 5 is not in B) and ends just before 8 (since 8 is not in B).

Therefore, A∩B = (5, 8). This is because the interval (5, 8) is the region where both sets overlap. Make sure you understand why we use parentheses here – neither 5 nor 8 are included in the intersection because they are not included in set B.

In essence, A∩B finds the area where both sets “overlap.” This concept is fundamental in set theory and has many applications in areas like logic and computer science.

A\B: The Set Difference (A minus B)

Now we're getting into set differences! A\B (sometimes written as A – B) means we want all the elements that are in set A but not in set B. Think of it as taking set A and removing anything that's also in set B.

So, we start with A = (4, 10] and we want to remove B = (5, 8). Visually, imagine erasing the portion of set A that overlaps with set B. This leaves us with two separate intervals:

  1. The part of A before B starts: (4, 5]. We include 5 here because 5 is not in B, so it stays in A\B.
  2. The part of A after B ends: [8, 10]. We include 8 here because 8 is not in B, and we include 10 because it was in A to begin with.

Therefore, A\B = (4, 5] ∪ [8, 10]. We use the union symbol ∪ because we have two separate intervals. This is a key point to remember when dealing with set differences!

To simplify, think of A\B as “what’s left of A after we take away B.” This is a very practical way to visualize and understand set difference.

B\A: The Set Difference (B minus A)

Finally, let's tackle B\A. This time, we want all the elements that are in set B but not in set A. It's the reverse of what we just did.

We start with B = (5, 8) and we want to remove anything that's also in A. However, if you look at our sets, B is entirely contained within A. This means that if we remove the part of B that's also in A, we're removing the entire set B!

Therefore, B\A = ∅ (the empty set). There are no elements in B that are not also in A. The empty set is a crucial concept in set theory, representing a set with no elements.

In simple words, since B is completely inside A, there’s nothing left in B after we remove the part that’s in A. This result highlights the importance of understanding the relationships between sets.

Graphical Representation

To really nail this down, let's visualize these operations on a number line. I'll describe it, and you can sketch it out on paper (or even use a digital drawing tool!).

  1. Draw a number line. Mark the key points: 4, 5, 8, and 10.
  2. Represent set A (4, 10] as a line segment. Use a parenthesis at 4 (open circle) and a square bracket at 10 (closed circle). Shade the line between 4 and 10.
  3. Represent set B (5, 8) as another line segment. Use parentheses at both 5 and 8 (open circles). Shade this line segment, perhaps using a different color.

Now, you can visually see:

  • A∪B is the combined shaded area from just above 4 to 10 (inclusive).
  • A∩B is the overlapping shaded area between 5 and 8.
  • A\B is the shaded area of A that doesn't overlap with B. You'll see two separate segments: one from just above 4 to 5 (inclusive) and another from 8 (inclusive) to 10 (inclusive).
  • B\A has no shaded area, visually representing the empty set.

A visual representation makes these concepts so much clearer! It's a fantastic way to double-check your calculations and understanding.

Key Takeaways

Let's recap what we've learned today:

  • Union (A∪B): Combines all elements from both sets.
  • Intersection (A∩B): Includes only the elements common to both sets.
  • Set Difference (A\B): Includes elements in A but not in B.
  • Set Difference (B\A): Includes elements in B but not in A.
  • Graphical Representation: Visualizing sets on a number line can significantly aid understanding.

Understanding these set operations is crucial for many areas of mathematics, computer science, and logic. The more you practice, the more natural these concepts will become.

So, there you have it! We've successfully calculated A∪B, A∩B, A\B, and B\A for the given sets and visualized them graphically. I hope this explanation was clear and helpful. Keep practicing, and you'll become a set theory pro in no time! Remember, understanding the definitions and visualizing the sets are the keys to success.