Interval Operations: Solve & Choose Correct Answer (A-D)

Hey guys! Let's dive into the fascinating world of interval operations! We're going to tackle some problems where we need to find the intersection and union of different intervals. It might sound a bit intimidating at first, but trust me, with a little practice, you'll become a pro at solving these. We'll break down each problem step-by-step and make sure you understand exactly what's going on. So, grab your thinking caps, and let's get started!

Understanding Interval Notation

Before we jump into solving problems, let's quickly review what interval notation actually means. Intervals are a way of representing a set of numbers that fall between two endpoints. We use brackets and parentheses to indicate whether the endpoints are included in the set or not. This is crucial for getting the right answers, so pay close attention, okay?

• Square brackets [ ]: These guys mean that the endpoint is included in the interval. For example, [a, b] means all the numbers between a and b, including a and b themselves.
• Parentheses ( ): These mean that the endpoint is not included. So, (a, b) means all the numbers between a and b, but excluding a and b.
• Infinity ∞: Infinity is a concept, not a number, so we always use parentheses with it. [a, ∞) means all numbers greater than or equal to a, stretching on forever in the positive direction. Similarly, (-∞, a] means all numbers less than or equal to a, going all the way to negative infinity.

Visualizing these intervals on a number line can be super helpful. Imagine a number line stretching from negative infinity to positive infinity. When you see an interval, think of it as a segment on that line. Square brackets mean a solid, filled-in endpoint, while parentheses mean an open, hollow endpoint. Got it? Great! Now, let's talk about the operations we'll be performing.

Intersection (∩) and Union (∪)

Okay, so now that we understand intervals, let's talk about the two main operations we'll be using: intersection and union. These are set operations, but when applied to intervals, they have specific meanings in terms of the numbers they include. Understanding these operations is key to solving our problems!

• Intersection (∩): The intersection of two intervals is the set of all numbers that are in both intervals. Think of it as the overlapping region. If you visualize the intervals on a number line, the intersection is the part where they overlap. It's like finding the common ground between two groups. To find the intersection, you need to identify the range of values that satisfy both interval conditions. This often involves finding the larger of the two lower bounds and the smaller of the two upper bounds. For instance, if one interval is all numbers greater than 2 and another is all numbers less than 5, the intersection is the set of numbers between 2 and 5. Remember, the endpoints are included in the intersection only if they are included in both original intervals. If one interval has a parenthesis at an endpoint and the other has a bracket, the intersection will have a parenthesis at that endpoint.
• Union (∪): The union of two intervals is the set of all numbers that are in either interval (or both!). Think of it as combining the intervals into one big set. On a number line, the union is the entire region covered by both intervals. It's like merging two groups together. Finding the union involves identifying the overall range of values covered by the intervals. You typically take the smaller of the two lower bounds and the larger of the two upper bounds. For example, if one interval includes numbers less than 0 and another includes numbers greater than 3, the union includes all numbers less than 0 as well as all numbers greater than 3. Be careful, though! If there's a gap between the intervals, the union will consist of separate intervals. In those cases, you'll express the union using the union symbol (∪) between the individual intervals.

To really nail this down, let's use a simple example. Consider the intervals [1, 3] and [2, 4]. The intersection, [1, 3] ∩ [2, 4], is [2, 3] because those are the numbers they both share. The union, [1, 3] ∪ [2, 4], is [1, 4] because it combines everything from both intervals.

Now that we've got the basics down, let's tackle the actual problems!

Solving the Interval Operations

Alright, guys, let's get our hands dirty and solve those interval problems! Remember our goal: to understand how to combine intervals using intersection and union. We'll go through each one step-by-step, visualizing the intervals and carefully considering the endpoints.

Problem a) [0, ∞) ∩ [-√2, 0) = ?

Okay, first up, we have the intersection of the interval [0, ∞) and the interval [-√2, 0). Let's break this down. The interval [0, ∞) represents all real numbers greater than or equal to 0. It starts at 0 (included because of the square bracket) and goes all the way to positive infinity. On a number line, you'd picture a line starting at 0 and extending infinitely to the right.

Now, let's look at the second interval, [-√2, 0). This represents all real numbers between -√2 and 0, including -√2 (square bracket) but excluding 0 (parenthesis). So, on the number line, it's a segment starting at -√2 and going up to, but not including, 0.

To find the intersection, we need to identify the numbers that are common to both intervals. Visually, where do these two segments overlap? They both include numbers between -√2 and 0. However, a crucial detail is that the interval [-√2, 0) does not include 0. Therefore, even though [0, ∞) includes 0, the intersection cannot. Thus, the intersection is [-√2, 0). This is the set of all numbers greater than or equal to -√2 but strictly less than 0.

Therefore, the correct answer is [-√2, 0). This is because the first interval includes all numbers from 0 to infinity, and the second interval includes all numbers from -√2 up to (but not including) 0. The only numbers they both have are those between -√2 (inclusive) and 0 (exclusive).

Problem b) (-∞, √2] ∪ (-√2, ∞) = ?

Next up, we have the union of (-∞, √2] and (-√2, ∞). Remember, the union means we're combining all the numbers from both intervals into one big set.

Let's start with (-∞, √2]. This interval includes all real numbers less than or equal to √2. On the number line, it's a line extending infinitely to the left, ending at √2 (included).

Now consider (-√2, ∞). This interval includes all real numbers strictly greater than -√2, going all the way to positive infinity. On the number line, it's a line starting just to the right of -√2 (not including -√2) and extending infinitely to the right.

When we take the union, we combine everything. The first interval covers everything to the left of √2, and the second interval covers everything to the right of -√2. The key thing to notice here is that there's no gap! Even though the second interval doesn't include -√2, the first interval does. And even though the first interval only goes up to √2, the second interval covers everything beyond √2. So, together, they cover all real numbers.

Therefore, the union of these two intervals is the set of all real numbers, which is represented as (-∞, ∞) or simply ℝ. The correct answer here is the set of all real numbers, because the intervals together cover the entire number line. Think of it as covering every possible value - there's no number left out!

Problem c) (-∞, √2] ∩ (-√2, ∞) = ?

Finally, let's tackle the intersection of (-∞, √2] and (-√2, ∞). Remember, the intersection is where the intervals overlap.

We already know what these intervals look like from the previous problem. (-∞, √2] includes everything less than or equal to √2, and (-√2, ∞) includes everything strictly greater than -√2. The question now is, what do they both include?

Visualize the number line. The first interval goes from negative infinity up to √2, including √2. The second interval goes from -√2 (not including -√2) to positive infinity. Where do these overlap? They overlap between -√2 and √2. But we have to be careful about the endpoints.

The first interval includes √2, and the second interval goes up to √2. So, √2 is in the intersection. However, the second interval does not include -√2. Even though the first interval includes -√2, the intersection only includes numbers present in both intervals. Therefore, -√2 is not in the intersection.

So, the intersection is the interval (-√2, √2]. This means all numbers greater than -√2 (not including -√2) and less than or equal to √2. The correct answer is (-√2, √2]. It's crucial to pay attention to whether the endpoints are included or excluded when dealing with intersections, guys!

Key Takeaways and Tips

Wow, we've covered a lot! Let's recap the most important things to remember when working with interval operations:

• Visualize on a number line: Drawing a number line and shading the intervals can make it much easier to see the intersection and union.
• Pay attention to endpoints: Square brackets mean the endpoint is included, parentheses mean it's excluded. This is super important for getting the right answer!
• Intersection is the overlap: It's the set of numbers common to both intervals.
• Union is the combination: It's the set of all numbers in either interval (or both).
• Real numbers (ℝ): Remember that (-∞, ∞) represents all real numbers.

Practice Makes Perfect!

Guys, the best way to master interval operations is to practice, practice, practice! Try solving similar problems on your own, and don't be afraid to make mistakes – that's how we learn! Review your solutions, understand where you went wrong, and keep at it. You've got this!

Interval operations might seem tricky at first, but with a solid understanding of the concepts and some diligent practice, you'll be solving them like a pro in no time. Just remember to visualize those number lines, pay close attention to the endpoints, and keep those key definitions of intersection and union fresh in your mind. Happy problem-solving!