Mastering Set Operations: Union & Intersection Explained

What's up, math whizzes! Today, we're diving deep into the super cool world of set theory. Specifically, we're going to tackle a problem that involves finding the elements of two sets, A and B, and then figuring out what happens when we combine them (union) and find what they have in common (intersection). This might sound a bit abstract at first, but trust me, once we break it down, it's going to click! We'll be working with two sets: Set A, which is all about two-digit numbers greater than 5, and Set B, which includes single-digit numbers greater than 7, and two-digit numbers that are bigger than 36 but smaller than 42. Get ready to flex those mathematical muscles because by the end of this, you'll be a pro at understanding how sets interact!

Understanding Set A: Two-Digit Numbers Greater Than 5

Alright guys, let's first focus on Set A. The definition here is pretty straightforward: Set A contains all two-digit numbers that are greater than 5. Now, when we talk about two-digit numbers, we're talking about integers ranging from 10 all the way up to 99. The condition is that these numbers must be greater than 5. Since all two-digit numbers inherently start from 10, which is already greater than 5, this condition actually means we just need to list all possible two-digit numbers. So, if you think about it, the smallest two-digit number is 10, and the largest is 99. Therefore, Set A is the collection of all integers from 10 to 99, inclusive. We can write this out formally as: A = {10, 11, 12, ..., 98, 99}. It's a pretty large set, right? We're not going to list every single number here, but it's important to understand the range. The key takeaway is that any number with two digits will automatically satisfy the 'greater than 5' condition. This makes defining Set A super simple. We just need to remember the definition of a two-digit number. So, when we are asked to find elements of Set A, we are essentially looking for any integer $x$ such that $10 \le x \le 99$. The problem statement might seem like it's adding an extra layer with 'greater than 5', but in the context of two-digit numbers, it's redundant information that doesn't change the set's composition. This is a common trick in math problems – to see if you can identify which conditions are actually restrictive and which are already met by other definitions. So, for Set A, we're dealing with the entire universe of two-digit integers. Pretty neat, huh?

Decoding Set B: Single Digits and Specific Two-Digit Numbers

Now, let's move on to Set B. This one has a couple of parts, so we need to pay close attention. Set B is defined as the collection of single-digit numbers that are greater than 7, and two-digit numbers that are greater than 36 and less than 42. Let's break this down. First, the single-digit numbers. Single-digit numbers are integers from 0 to 9. We need the ones that are greater than 7. Out of 0 through 9, only one number fits this bill: 8. So, that's the first part of Set B. Next, we have the two-digit numbers. The condition here is that they must be greater than 36 and less than 42. Let's list the integers that fall within this range: 37, 38, 39, 40, 41. All of these are two-digit numbers, and they all satisfy both conditions (greater than 36 and less than 42). So, we combine these elements. Therefore, Set B consists of the number 8, and the numbers 37, 38, 39, 40, and 41. We can write this out formally as: B = {8, 37, 38, 39, 40, 41}. Notice how Set B is a much smaller and more specific collection compared to Set A. It's important to be precise when defining the elements of a set. In this case, we have a mix of single-digit and two-digit numbers, each with its own set of criteria. The logical connector 'and' in the definition means both conditions must be met for a number to be included in Set B. So, a number like 9 (single-digit but not greater than 7) is out, and a number like 36 (two-digit but not greater than 36) or 42 (two-digit but not less than 42) are also excluded. This careful consideration of the boundaries and conditions is crucial for correctly identifying all members of Set B. We've got our elements for Set B sorted!

Finding the Union of Sets A and B (A ∪ B)

Alright, folks, now that we've got our sets defined, it's time to perform some operations! First up, we're looking for the union of Sets A and B, denoted as $A \cup B$. The union of two sets is essentially a new set that contains all the elements from both original sets, with no duplicates. Think of it like merging two groups of friends; everyone from both groups is now in one big supergroup. So, we need to take all the numbers from Set A and all the numbers from Set B and put them together. Remember, Set A is 10, 11, 12, ..., 98, 99}, and Set B is {8, 37, 38, 39, 40, 41}. When we combine them, we start by listing all elements of A. Then, we add any elements from B that are not already in A. Let's look at Set B's elements 8 is a single-digit number, so it's definitely not in Set A (which only contains two-digit numbers). The numbers 37, 38, 39, 40, and 41 are all two-digit numbers. Are they already in Set A? Yes, they are, because Set A includes all two-digit numbers from 10 to 99. So, while these numbers are present in Set B, they are also present in Set A. When we form the union, we don't list them twice. Therefore, the union $A \cup B$ will contain all the elements of A, plus the element 8 from Set B (which wasn't in A). So, $A \cup B$ = {8, 10, 11, 12, ..., 36, 37, 38, 39, 40, 41, ..., 98, 99. It's basically Set A with the single-digit number 8 added to it. The 'union' operation is all about inclusion – bringing everything together. We list every unique element that appears in either set or both. Since Set A already covers all numbers from 10 to 99, and Set B includes 8 and some numbers within the 10-99 range, the union is effectively all numbers from 10 to 99, plus the number 8. It's a comprehensive collection that captures every number present in either of our initial sets. It's like saying, "Give me everything from A, and give me everything from B, but don't give me any duplicates." This ensures that each distinct number is represented exactly once in the final union set. Pretty straightforward once you get the hang of it!

Determining the Intersection of Sets A and B (A ∩ B)

Finally, let's dive into the intersection of Sets A and B, denoted as $A \cap B$. The intersection is the opposite of the union; it's the set that contains only the elements that are common to both Set A and Set B. Think of it as finding the overlap between two circles in a Venn diagram. We need to look at the elements of Set A and Set B and identify which numbers appear in both. Set A = 10, 11, 12, ..., 98, 99} and Set B = {8, 37, 38, 39, 40, 41}. Let's go through the elements of Set B and see if they are in Set A. The first element of Set B is 8. Is 8 in Set A? No, because Set A only contains two-digit numbers, and 8 is a single-digit number. So, 8 is not in the intersection. Now, let's look at the other elements of Set B 37, 38, 39, 40, and 41. Are these numbers in Set A? Yes, they are! Set A contains all two-digit numbers from 10 to 99, and 37, 38, 39, 40, and 41 all fall within this range. Since these numbers are present in both Set A and Set B, they are part of the intersection. So, the intersection $A \cap B$ is the set containing these common elements: $A \cap B$ = {37, 38, 39, 40, 41. This means these specific numbers are the only ones that satisfy the conditions for being in both Set A and Set B. The intersection operation is all about exclusivity – finding what's shared. It highlights the elements that are simultaneously members of both sets. This is incredibly useful in many areas of math and computer science, such as database querying or logic puzzles, where you need to find commonalities. So, we've successfully identified the elements that belong to both sets, which are the numbers from 37 to 41. Keep practicing these operations, and you'll master set theory in no time!