Set Theory: Understanding Union, Subsets, And Proper Subsets

Hey math enthusiasts! Let's dive into some fundamental concepts in set theory. This stuff is super important for building a solid foundation in mathematics. We're going to explore the ideas of set union, subsets, and proper subsets, using the sets A and B provided in the problem. By the end, you'll be able to confidently determine the correct statement and understand the underlying principles. Ready to get started? Let's go!

Unveiling the Union of Sets: Understanding A ∪ B

First up, let's look at the concept of the union of sets. The union of two sets, often denoted by the symbol '∪', is a new set that contains all the elements that are in either set, or both sets. Basically, you combine all the unique elements from both sets into a single, combined set. So, if we are given set A = {4, 7, 10, 13, 17} and set B = {3, 5, 7, 9}, the union, A ∪ B, would include all the numbers present in either A, B, or both. The important thing is that each element only appears once in the resulting set, even if it's present in both A and B. When you combine them, you only list each unique number once, avoiding any repeats. Thus, A ∪ B becomes the set that holds all elements from set A and set B. The key is to grab every single unique number from each set and combine them into one new set. It's like a big mathematical mix-and-match, where you gather every single element, ensuring no repetitions, and assemble them into a brand-new set.

So, if we apply this to our problem, we're looking to form the union of sets A and B. Set A has the elements 4, 7, 10, 13, and 17. Set B has the elements 3, 5, 7, and 9. Now, to find A ∪ B, we're going to combine these two sets and make sure there are no duplicate entries. Both sets contain the number 7, but since we only need to list each element once, we include it just once in the union. To form A ∪ B, we list all the elements from both sets: 3, 4, 5, 7, 9, 10, 13, and 17. Hence, A ∪ B = {3, 4, 5, 7, 9, 10, 13, 17}. So, the option which states A ∪ B = {7} is completely incorrect because it only lists one element, 7, and it misses all the other numbers that are part of the union. The correct union of A and B includes every single unique element found in both sets, which the option A ∪ B = {7} clearly does not do. Thus, statement A is false. The union operation essentially gives you a broader set that includes every single element in either or both of the original sets without any duplicates, making it a very simple yet important operation in set theory.

Deciphering Subsets: Is B ⊆ B?

Next, let's explore subsets. A set is a subset of another set if all of its elements are also elements of the other set. The symbol '⊆' is used to denote a subset. In other words, if every single element of set B is also found within set A, then we can say that B is a subset of A. Now, a critical thing to remember is that every set is a subset of itself. This might seem a little weird at first, but it makes sense when you think about the definition. Because all the elements of B are, of course, in B itself, B is, by definition, a subset of B. The statement B ⊆ B is always true. It's a fundamental property of sets. It says that the set B is a subset of itself, which, based on the definition of a subset, is always correct. Every set contains all of its own elements. Think of it like this: If you have a box (set B), it definitely contains everything that’s already inside the same box (set B). It doesn’t matter what the elements are, if you are looking at B in comparison to itself, you will always find that B is a subset of B. It’s like saying, “Everything in this room is also in this room.” It's a basic, fundamental idea, and it's something that is always true in the world of sets. The statement B ⊆ B is always true. So, based on the definition, B is indeed a subset of itself. Option B correctly states B ⊆ B, which is always true because any set is a subset of itself. That's a core property of sets.

Now, because B is a subset of itself, and option B tells us precisely that, it is definitely a true statement. It's like saying that everything in the set is also in the set. This simple idea helps form a strong base for understanding more complex set-related concepts. So, option B is correct because the property B ⊆ B holds true based on the definition of a subset, where every set is considered a subset of itself. Therefore, the statement is valid and always true. It is a fundamental property of set theory. Given that all elements of a set are contained within the set itself, the statement that B is a subset of B is undeniably true. So, the correct answer must be B.

Differentiating Proper Subsets: Is B ⊂ A?

Finally, let's tackle proper subsets. A proper subset, denoted by '⊂', is a subset that is not equal to the original set. This means that all the elements of a proper subset are in the original set, but the original set must also have at least one element that isn't in the proper subset. In simpler terms, a proper subset is a smaller version of a set, entirely contained within a larger set. To determine if B is a proper subset of A (B ⊂ A), we need to see if all elements of B are in A, and if A has at least one element that is not in B. Now, let’s check if B is a proper subset of A. Set B = {3, 5, 7, 9} and Set A = {4, 7, 10, 13, 17}. Does every element in B also exist in A? Let's check: 3 is in B but not in A. 5 is in B but not in A. 7 is in both B and A. 9 is in B but not in A. Because not all elements of B are in A, B is not a subset of A, let alone a proper subset. B contains elements that A does not. Therefore, B cannot be a proper subset of A. To be a proper subset, all elements of B must be in A, and A must have at least one element not in B. In this case, since B contains elements not in A, then B is not even a subset of A, thus it cannot be a proper subset. The idea of a proper subset means that the entire set B should be contained within set A, with set A having at least one element that set B doesn’t contain. So, since not all the elements of B can be found in set A, B is not a subset of A, so the proper subset relationship isn’t possible. This means that statement C is false.

Looking at this closely, for B to be a proper subset of A, every element in B must be present in A, but A should have at least one element not present in B. However, as we have already seen, elements like 3, 5 and 9 are present in set B but not in set A. This immediately tells us that the statement B ⊂ A is false. The definition of a proper subset requires that all of B's elements are also in A, which is not the case here. Therefore, option C is incorrect because B is not a proper subset of A, thus the statement is false. The difference between a subset and a proper subset is that a proper subset has to be strictly smaller than the original set. Since B has elements that aren’t in A, it cannot be a proper subset of A.

Conclusion: Selecting the Correct Statement

Alright, let’s wrap this up, guys! We have gone through each of the options, carefully dissecting each statement using the fundamental principles of set theory. We have learned about the union of sets, and the definition of subsets and proper subsets. We have identified that the union of A and B is not equal to {7}, and we've established that B is a subset of itself (B ⊆ B). We also found that B cannot be a proper subset of A because B has elements that aren’t in A. So, after a thorough review of the statements, the correct statement is B ⊆ B. Remember, understanding these basic concepts is crucial as you continue to explore more advanced mathematical ideas. Keep practicing and keep up the great work!