Determine Sets: Examples And Explanations

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Hey guys! Today, we're diving into the fascinating world of set theory and tackling the problem of determining sets based on different conditions. Set theory is a fundamental concept in mathematics, and understanding how to define and work with sets is crucial for various fields, including computer science, statistics, and more. We'll break down each example step-by-step, so you can confidently handle similar problems in the future.

a) (3,9) ∩ Z

Let's kick things off with our first set: (3,9) ∩ Z. This notation involves the intersection of two sets: an open interval (3,9) and the set of integers (Z). Understanding what each part represents is key to finding the solution.

  • (3,9): This represents an open interval on the real number line. An open interval excludes its endpoints, meaning it includes all numbers strictly between 3 and 9, but not 3 and 9 themselves. Think of it as all numbers greater than 3 and less than 9. We can visualize this on a number line where we use parentheses to indicate the exclusion of the endpoints.
  • Z: This is the standard symbol for the set of integers. Integers are whole numbers (no fractions or decimals) and can be positive, negative, or zero. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.
  • ∩: This symbol represents the intersection of sets. The intersection of two sets is a new set containing only the elements that are common to both original sets. In simpler terms, it's what the sets have in common.

So, to determine (3,9) ∩ Z, we need to find the integers that lie strictly between 3 and 9. Let's list the integers greater than 3: 4, 5, 6, 7, 8, 9, 10... However, we only want the integers less than 9. Therefore, the integers that fit our criteria are 4, 5, 6, 7, and 8.

Thus, the solution to (3,9) ∩ Z is the set {4, 5, 6, 7, 8}. This set contains all the integers that are simultaneously within the open interval (3,9).

In conclusion, understanding the notation is paramount. Open intervals, sets of integers, and the intersection symbol are fundamental concepts. Visualizing the problem on a number line can often help to clarify the solution. Carefully listing the elements that meet the conditions ensures accuracy.

b) (-4, 5] ∩ N

Now, let's move on to our second example: (-4, 5] ∩ N. This problem also involves finding the intersection of two sets, but this time we have a slightly different interval and a different set to consider. Let's break it down:

  • (-4, 5]: This represents a half-open (or half-closed) interval. It includes all real numbers greater than -4 and less than or equal to 5. The parenthesis on the -4 side indicates that -4 is not included, while the square bracket on the 5 side means that 5 is included. So, we're looking at numbers from just above -4 up to and including 5.
  • N: This symbol represents the set of natural numbers. Natural numbers are positive integers, starting from 1. The set N is typically defined as {1, 2, 3, 4, ...}. Zero is sometimes included in the set of natural numbers depending on the convention, but in this context, we'll assume it starts from 1.
  • ∩: As before, this symbol represents the intersection of the sets. We're looking for the elements that are present in both the interval (-4, 5] and the set of natural numbers N.

To determine (-4, 5] ∩ N, we need to find the natural numbers that fall within the interval (-4, 5]. Natural numbers are positive integers, so we're looking for integers greater than 0 that are also less than or equal to 5. This means we consider the integers 1, 2, 3, 4, and 5.

Since all of these integers are within the interval (-4, 5], the solution to (-4, 5] ∩ N is the set {1, 2, 3, 4, 5}. This set contains all the natural numbers that are also within the specified interval.

The key takeaway here is understanding the difference between open, closed, and half-open intervals. The inclusion or exclusion of the endpoints is crucial. Knowing the definition of natural numbers is also essential. We are only considering the positive integers. Remember to pay close attention to the symbols used to represent different sets and intervals to avoid mistakes.

c) [-√13, √5] ∩ Z

Let's tackle our third example: [-√13, √5] ∩ Z. This one involves square roots, which might seem a bit intimidating, but we'll break it down into manageable parts. We're still dealing with the intersection of a set with the integers, but now our interval involves irrational numbers.

  • [-√13, √5]: This represents a closed interval. It includes all real numbers between -√13 and √5, including the endpoints. The square brackets indicate that both -√13 and √5 are part of the interval. To understand the interval better, we need to approximate the values of the square roots.
    • √13 is between √9 (which is 3) and √16 (which is 4). A closer approximation is that √13 is approximately 3.6.
    • √5 is between √4 (which is 2) and √9 (which is 3). A closer approximation is that √5 is approximately 2.2.
    • Therefore, our interval is approximately [-3.6, 2.2].
  • Z: As before, this is the set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
  • ∩: The intersection symbol means we need to find the integers that are within the interval [-√13, √5].

To determine [-√13, √5] ∩ Z, we need to identify the integers that lie between -√13 (approximately -3.6) and √5 (approximately 2.2), including the endpoints (if they were integers). Let's list the integers within this range:

  • -3 is greater than -3.6, so it's included.
  • -2 is within the range.
  • -1 is within the range.
  • 0 is within the range.
  • 1 is within the range.
  • 2 is less than 2.2, so it's included.
  • 3 is greater than 2.2, so it's excluded.

Therefore, the solution to [-√13, √5] ∩ Z is the set {-3, -2, -1, 0, 1, 2}. These are all the integers that fall within the closed interval defined by the square roots.

This example highlights the importance of approximating irrational numbers to better understand their position on the number line. Carefully consider the endpoints of the interval and whether they are included or excluded. And systematically listing the integers within the range helps to avoid errors.

d) R \ (0, ∞)

Finally, let's tackle our last example: R \ (0, ∞). This one introduces a new symbol and a different type of operation on sets. Instead of intersection, we're dealing with the set difference.

  • R: This symbol represents the set of all real numbers. Real numbers include all rational numbers (like integers, fractions, and terminating or repeating decimals) and irrational numbers (like √2 and π). It's essentially the entire number line.
  • (0, ∞): This is an open interval representing all real numbers greater than 0. The parenthesis indicates that 0 is not included, and ∞ represents infinity, meaning the interval extends indefinitely in the positive direction.
  • : This symbol represents the set difference (also sometimes written as a minus sign, -). The set difference A \ B (read as