Set Operations: B ∩ C And C - A Solved!
Hey everyone! Today, let's dive into a fun math problem involving sets and set operations. We're given four sets: A, B, C, and D, and our mission is to find the intersection of sets B and C (B ∩ C), as well as the difference between set C and set A (C - A). Sounds exciting, right? Let's break it down step by step.
Understanding the Basics: Sets and Set Operations
Before we jump into the solution, let's quickly recap what sets and set operations are. Think of a set as a collection of distinct objects. These objects can be anything – numbers, letters, even other sets! In our case, sets A, B, C, and D contain pairs of numbers, which we'll treat as coordinates (more on that later).
Now, let's talk about the operations we'll be using:
- Intersection (∩): The intersection of two sets is a new set containing only the elements that are present in both original sets. Imagine it like finding the common ground between two groups. If an element is not in both sets, it doesn't make the cut for the intersection.
- Difference (-): The difference between two sets (A - B) is a new set containing elements that are present in set A but not in set B. It's like taking set A and removing anything that's also in set B. Only the unique elements of A remain.
With these definitions in mind, we're well-equipped to tackle the problem.
Defining Our Sets: A, B, C, and D
Okay, let's get down to business and define our sets explicitly. We're given the following:
- Set A = { [3, 7] }
- Set B = { [-2, 4] }
- Set C = { [-5, 1] }
- Set D = { [-1, 6] }
Notice that each set contains a single element, which is a pair of numbers enclosed in square brackets. We can interpret these pairs as coordinates in a two-dimensional plane (x, y). This is a crucial observation because it allows us to visualize these sets geometrically, which can be super helpful in understanding set operations.
Think of set A as a single point at coordinates (3, 7), set B as a point at (-2, 4), set C as a point at (-5, 1), and set D as a point at (-1, 6). They're just four isolated points in space!
Solving for B ∩ C (Intersection of B and C)
Alright, the first part of our mission is to find the intersection of sets B and C (B ∩ C). Remember, the intersection contains elements that are present in both B and C. Let's take a look:
- Set B = { [-2, 4] }
- Set C = { [-5, 1] }
Do you see any elements that are common to both sets? Nope! Set B contains the coordinate pair [-2, 4], while set C contains [-5, 1]. These are distinct points, and there's no overlap. Therefore, the intersection of B and C is an empty set. An empty set is a set that contains no elements, and it's often denoted by the symbol Ø or {}.
So, we can confidently say:
B ∩ C = Ø
In simpler terms, there are no points that belong to both set B and set C. They're like two separate islands, with no shared territory.
Solving for C - A (Difference between C and A)
Now, let's move on to the second part: finding the difference between set C and set A (C - A). Remember, the difference C - A contains elements that are in C but not in A. Let's examine the sets again:
- Set C = { [-5, 1] }
- Set A = { [3, 7] }
We need to identify elements that are in C but not in A. Set C contains the coordinate pair [-5, 1], and set A contains [3, 7]. Are these the same? Absolutely not! The point [-5, 1] is unique to set C; it's not present in set A. Therefore, when we take the difference C - A, we're left with the single element that was originally in C.
So, we can conclude:
C - A = { [-5, 1] }
In plain language, set C minus set A is simply set C itself. This makes sense because A and C don't share any elements, so subtracting A from C doesn't change C.
Visualizing the Sets and Operations
To solidify our understanding, let's visualize these sets and operations on a coordinate plane. Imagine a graph with an x-axis and a y-axis. We can plot our points:
- A: (3, 7)
- B: (-2, 4)
- C: (-5, 1)
- D: (-1, 6)
You'll see four distinct points scattered on the plane. The intersection of B and C (B ∩ C) is empty because there's no overlap between the points representing B and C. The difference C - A (C - A) is just the point C itself because subtracting A doesn't remove anything from C.
Key Takeaways and Why This Matters
Okay, guys, we've successfully solved for the intersection of B and C and the difference between C and A! Let's recap the key takeaways:
- B ∩ C = Ø (The intersection of B and C is an empty set).
- C - A = { [-5, 1] } (The difference between C and A is set C itself).
Now, you might be wondering,
