Define Sets: A To F | Exercise 99 Math Solution

Hey guys! Today, we're diving deep into Exercise 99, which involves defining several sets explicitly. This is a common type of problem in mathematics, especially when you're dealing with real numbers and inequalities. So, let's break it down step by step, making sure everyone understands the process. We'll tackle each set individually, giving you a clear explanation along the way.

a) Defining the Set A = {x ∈ R / x^2 ≤ 4}

Okay, let's start with set A, which is defined as all real numbers (x) such that x squared is less than or equal to 4. This is where understanding inequalities becomes super important. To explicitly define this set, we need to figure out the range of x values that satisfy the condition x^2 ≤ 4. Think of it like finding the sweet spot – the numbers that, when squared, don't exceed 4.

Here’s how we can approach it:

First, consider the equation x^2 = 4. The solutions to this equation are x = 2 and x = -2. These are our boundary points. They tell us where the set potentially starts and stops. But remember, we're dealing with an inequality (≤), not just an equality (=). This means we're looking for a range of values, not just two specific numbers.

Now, we need to figure out what happens in between and outside these boundary points. A great way to do this is by testing some values. Let’s try x = 0, which is between -2 and 2. If we plug 0 into x^2 ≤ 4, we get 0^2 ≤ 4, which simplifies to 0 ≤ 4. This is true, so the values between -2 and 2 are part of our set. This is a crucial step to visualize the solution set on the number line.

Next, let’s test a value outside this range, say x = 3. Plugging this in, we get 3^2 ≤ 4, which simplifies to 9 ≤ 4. This is false. So, values greater than 2 are not part of the set. Similarly, if we test x = -3, we get (-3)^2 ≤ 4, which simplifies to 9 ≤ 4, also false. So, values less than -2 are not in the set either. Testing these values on the number line is a valuable technique for understanding inequalities.

Therefore, the set A includes all real numbers between -2 and 2, inclusive (because of the “equal to” part in ≤). In interval notation, we write this as A = [-2, 2]. This means that -2 and 2 themselves are also part of the set.

In summary, set A is the closed interval from -2 to 2. That wasn't so bad, right? Let's keep going! Understanding the concept of closed intervals is vital here, as it indicates that the endpoints are included in the set.

b) Defining the Set B = {x ∈ R / x^2 < 9}

Alright, let's tackle set B! This one is quite similar to set A, but with a slight twist. Here, we're looking for all real numbers (x) such that x squared is less than 9. Notice the difference? We have a strict inequality (<) this time, which means the boundary points won't be included in the set. Understanding the nuances of strict inequalities is key to defining the set accurately.

Let's break it down:

Just like before, we start by considering the equation x^2 = 9. The solutions here are x = 3 and x = -3. These are our new boundary points, but this time, they won't be part of the final set because we have a strict inequality. This distinction is critical when defining sets with strict inequalities.

Now, let's test some values. Again, x = 0 is a good choice since it's between -3 and 3. Plugging it into x^2 < 9, we get 0^2 < 9, which simplifies to 0 < 9. This is true, so the values between -3 and 3 are part of set B. Visualizing these values on a number line can enhance understanding.

What about values outside this range? If we try x = 4, we get 4^2 < 9, which simplifies to 16 < 9. This is false. Similarly, for x = -4, we get (-4)^2 < 9, which simplifies to 16 < 9, also false. So, values outside the range of -3 to 3 are not included. Testing these boundary conditions is essential for defining the set correctly.

Since we have a strict inequality, the boundary points -3 and 3 are not included in the set. Therefore, set B includes all real numbers strictly between -3 and 3. In interval notation, we write this as B = (-3, 3). The parentheses indicate that the endpoints are not included. This notation is standard for representing open intervals, where the boundaries are excluded.

So, set B is the open interval from -3 to 3. We're on a roll! Recognizing the difference between open and closed intervals is fundamental in set theory.

c) Defining the Set C = {x ∈ R / |x-3| ≤ 1}

Okay, this one looks a bit trickier, but don't worry, we've got this! Set C is defined as all real numbers (x) such that the absolute value of x minus 3 is less than or equal to 1. Absolute value inequalities might seem intimidating, but they're actually quite manageable once you understand the concept. The key here is to remember that absolute value represents the distance from zero. Interpreting absolute value in terms of distance is crucial for solving these inequalities.

Let's break down the absolute value inequality |x - 3| ≤ 1:

Remember that |x - 3| represents the distance between x and 3 on the number line. So, we're looking for all x values that are within a distance of 1 from 3. This is where the geometric interpretation of absolute value becomes handy.

This inequality is equivalent to two separate inequalities: -1 ≤ (x - 3) ≤ 1. We can solve this compound inequality by adding 3 to all parts:

-1 + 3 ≤ x - 3 + 3 ≤ 1 + 3

This simplifies to:

2 ≤ x ≤ 4

So, set C includes all real numbers between 2 and 4, inclusive. This means that both 2 and 4 are part of the set. Understanding the translation from absolute value inequality to a compound inequality is the core of solving these problems.

In interval notation, we write this as C = [2, 4]. This represents a closed interval, as both endpoints are included. Visualizing this interval on a number line can further clarify the solution.

Therefore, set C is the closed interval from 2 to 4. Great job! Understanding how to manipulate inequalities is a vital skill in mathematics.

d) Defining the Set D = {x ∈ R / x ≥ 2}

Now we're on to set D, which is defined as all real numbers (x) such that x is greater than or equal to 2. This one is pretty straightforward! We're looking for all numbers that are 2 or larger. This type of inequality represents a ray on the number line, extending infinitely in one direction. Understanding how to represent these sets is crucial for advanced mathematical concepts.

Let's express this in interval notation:

Since x can be 2 (because of the “equal to” part) and can be any number greater than 2, the set extends infinitely to the right. In interval notation, we write this as D = [2, ∞). The square bracket indicates that 2 is included, and the infinity symbol indicates that the set continues indefinitely in the positive direction. The use of the infinity symbol is a standard notation for unbounded intervals.

So, set D is the closed interval from 2 to infinity. Easy peasy! Familiarity with interval notation is essential for communicating mathematical concepts effectively.

e) Defining the Set E = {x ∈ R / 1 < x ≤ 5}

Set E is defined as all real numbers (x) such that x is greater than 1 and less than or equal to 5. This is another compound inequality, but this time, we have a strict inequality on one end and a non-strict inequality on the other. This combination results in a half-open interval, where one endpoint is included, and the other is excluded. Recognizing these mixed inequalities is important for correctly defining the set.

Let's express this in interval notation:

Since x is strictly greater than 1 (1 < x), 1 is not included in the set. However, x can be equal to 5 (x ≤ 5), so 5 is included. In interval notation, we write this as E = (1, 5]. The parenthesis on the left indicates that 1 is not included, and the square bracket on the right indicates that 5 is included. This mixed notation clearly conveys the boundaries of the set.

Therefore, set E is the half-open interval from 1 (exclusive) to 5 (inclusive). You're doing great! The ability to handle mixed inequalities is a valuable skill in mathematical analysis.

f) Defining the Set F = {x ∈ R / x^3 ≤ 1}

Last but not least, we have set F, which is defined as all real numbers (x) such that x cubed is less than or equal to 1. This one involves a cubic inequality, which might seem a bit different, but the underlying principles are the same. The key here is to consider the behavior of the cubic function and how it relates to the inequality. Understanding the properties of cubic functions is crucial for solving these types of problems.

Let's break it down:

To solve this inequality, we first consider the equation x^3 = 1. The real solution to this equation is x = 1. This is our boundary point. Now, we need to determine the range of values for which x^3 ≤ 1.

Unlike quadratic inequalities, cubic inequalities involving real numbers are a bit simpler because the cubic function is monotonically increasing. This means that if x is less than 1, then x^3 will also be less than 1. If x is greater than 1, then x^3 will be greater than 1. This monotonic property simplifies the analysis of cubic inequalities.

So, we're looking for all x values less than or equal to 1. In interval notation, we write this as F = (-∞, 1]. The parenthesis on the left indicates that the set extends infinitely in the negative direction, and the square bracket on the right indicates that 1 is included. This unbounded interval represents all real numbers less than or equal to 1.

Therefore, set F is the interval from negative infinity to 1 (inclusive). Awesome job! Understanding the behavior of different types of functions is essential in advanced mathematics.

Conclusion

Phew! We've made it through all the sets in Exercise 99. Hopefully, you now have a clearer understanding of how to define sets explicitly, especially when dealing with inequalities and absolute values. Remember, the key is to break down the problem into smaller parts, identify the boundary points, and test values to determine the range that satisfies the given condition. You guys rock!

Remember to practice these techniques with various examples. Keep honing your skills, and you'll become a set-defining pro in no time! If you have any questions, don't hesitate to ask. Keep exploring and keep learning! Until next time, happy set-defining! Understanding set notation and interval notation is a foundational skill for many areas of mathematics, so mastering these concepts is a worthwhile endeavor. Keep practicing, and you'll get the hang of it!