Cardinality Of Set A: X² < 10, X ∈ Z

Hey guys! Let's dive into this interesting math problem where we need to find the number of elements (cardinality) in a set. The set A is defined by a condition involving squares of integers. Understanding sets and their properties is super important in mathematics, especially when you're dealing with things like functions, relations, and even more advanced topics later on. So, let's break this down step by step and make sure we get a solid grasp of how to solve this kind of problem. We'll go through the definition of the set, figure out which integers fit the condition, and then count them up. Ready? Let's get started!

Understanding the Set Definition

First, let's really understand what the set A is all about. The set is defined as A = x x² < 10, x ∈ Z. This notation might seem a bit cryptic at first, but it's actually quite straightforward once you break it down. The curly braces {} mean we're talking about a set. Inside the braces, we have a description of the elements that belong to the set. The 'x:' part tells us that we're looking for elements 'x'. Now, the juicy part: 'x² < 10' is the condition that 'x' must satisfy to be included in the set. This means we're looking for numbers 'x' such that when you square them (multiply them by themselves), the result is less than 10. Finally, 'x ∈ Z' tells us that 'x' must be an integer. Remember, integers are whole numbers (no fractions or decimals), and they can be positive, negative, or zero. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.

So, putting it all together, set A consists of all integers 'x' whose square is less than 10. This is the key to solving the problem. To find the cardinality (the number of elements) of A, we need to identify all the integers that meet this condition. Think of it like a puzzle: we have a rule, and we need to find all the numbers that fit the rule. Now, let's start figuring out which integers actually belong in set A!

Identifying the Elements of Set A

Okay, now for the fun part: let's figure out exactly which integers fit the bill and belong in our set A! Remember, we need to find all integers 'x' such that x² < 10. A great way to tackle this is to test out some integers, both positive and negative, and see if their squares are less than 10. Let's start with some positive integers. If we try x = 1, then x² = 1² = 1, which is definitely less than 10. So, 1 is in our set! How about x = 2? Then x² = 2² = 4, which is also less than 10. So, 2 is in the set too. Let's keep going. If x = 3, then x² = 3² = 9. Bingo! 9 is less than 10, so 3 is also in our set A. But what happens when we try x = 4? Then x² = 4² = 16. Uh oh! 16 is not less than 10, so 4 is out. And any integer larger than 4 will also have a square greater than 10, so we don't need to check those.

Now, let's think about negative integers. What happens when we square a negative number? Remember, a negative times a negative is a positive! So, if we try x = -1, then x² = (-1)² = 1, which is less than 10. So, -1 is in the set. Similarly, if x = -2, then x² = (-2)² = 4, which is also less than 10. So, -2 is in the set. And if x = -3, then x² = (-3)² = 9, which is less than 10, so -3 belongs in A as well. If we try x = -4, we get x² = (-4)² = 16, which is too big, just like the positive 4. There's one more integer we haven't checked yet: zero! If x = 0, then x² = 0² = 0, which is definitely less than 10. So, 0 is also in our set. Now we've tested a bunch of integers and figured out exactly which ones satisfy our condition. Let's put them all together to see what our set A looks like.

Determining the Cardinality of A

Alright, we've done the hard work of identifying all the elements that belong in set A. Now comes the easy part: counting them! Let's list out all the integers we found that satisfy the condition x² < 10: -3, -2, -1, 0, 1, 2, and 3. So, set A is {-3, -2, -1, 0, 1, 2, 3}. To find the cardinality of A, which is written as s(A), we simply count how many elements are in the set. We have 1, 2, 3, 4, 5, 6, 7 elements. Therefore, the cardinality of set A, s(A), is 7. That's it! We've solved the problem. We figured out the definition of the set, identified all the elements that belong to it, and then counted them up to find the cardinality.

This is a great example of how careful thinking and step-by-step problem-solving can help you tackle math questions. Remember, understanding the definitions and conditions is crucial. And don't be afraid to test out some numbers to see what works! Now, you're well-equipped to handle similar problems involving sets and inequalities. Keep up the great work, and you'll be a math whiz in no time!

Conclusion

So, guys, we've successfully navigated this math problem! To recap, we were asked to find the cardinality (number of elements) of set A, where A = x x² < 10, x ∈ Z. This means we needed to find all integers 'x' whose square is less than 10. We systematically tested integers, both positive and negative, and zero, to see which ones fit the condition. We found that the integers -3, -2, -1, 0, 1, 2, and 3 all satisfied the condition. Therefore, set A is {-3, -2, -1, 0, 1, 2, 3}. Finally, we counted the elements in set A and found that there are 7 elements. Thus, the cardinality of set A, s(A), is 7. This problem highlights the importance of understanding set notation, inequalities, and the properties of integers. By breaking down the problem into smaller steps and carefully applying the definitions, we were able to arrive at the correct answer. Remember, practice makes perfect! The more you work with these types of problems, the more comfortable and confident you'll become. Keep exploring the world of math, and you'll discover even more interesting concepts and challenges. You got this!