Verifying The Associative Law Of Intersection For Sets X, Y, And Z

Hey guys! Today, we're diving into the fascinating world of set theory to verify the associative laws of union and intersection. We'll be working with three sets: X, Y, and Z. Our mission is to prove that the associative laws hold true for these sets. So, buckle up and let's get started!

Defining Our Sets

First, let's define the sets we'll be working with. We have:

• Set X: This set contains the numbers from 1 to 10. So, X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
• Set Y: This set consists of the even numbers 2, 4, 6, and 8, along with the number 7. Thus, Y = {2, 4, 6, 7, 8}.
• Set Z: This set includes the numbers 5, 6, 7, and 8. Therefore, Z = {5, 6, 7, 8}.

Now that we have our sets clearly defined, we can move on to understanding the associative laws and how to verify them.

Understanding Associative Laws

Before we jump into the verification, let's make sure we're all on the same page about what associative laws actually mean. In set theory, we have two primary associative laws:

1. Associative Law of Union: This law states that the way we group sets when performing the union operation doesn't change the result. In other words, if we have three sets, A, B, and C, then (A ∪ B) ∪ C is the same as A ∪ (B ∪ C). Union, denoted by '∪', means combining all the elements from the sets into one set, removing any duplicates.
2. Associative Law of Intersection: Similar to the union, this law states that the grouping of sets doesn't affect the outcome when performing the intersection operation. For sets A, B, and C, (A ∩ B) ∩ C is equivalent to A ∩ (B ∩ C). Intersection, denoted by '∩', means finding the elements that are common to all the sets involved.

With these definitions in mind, let's proceed to verify these laws for our sets X, Y, and Z.

Verifying the Associative Law of Intersection

Okay, guys, let's start with the associative law of intersection. We need to prove that:

X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z

To do this, we'll calculate both sides of the equation separately and then compare the results. If they're the same, we've successfully verified the law for these sets.

Step 1: Calculate Y ∩ Z

First, we need to find the intersection of sets Y and Z. Remember, intersection means the elements that are present in both sets. Looking at our sets:

• Y = {2, 4, 6, 7, 8}
• Z = {5, 6, 7, 8}

The elements common to both Y and Z are 6, 7, and 8. So:

Y ∩ Z = {6, 7, 8}

Step 2: Calculate X ∩ (Y ∩ Z)

Now that we have Y ∩ Z, we can find the intersection of X with this result. Our sets are:

• X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• Y ∩ Z = {6, 7, 8}

The elements common to both X and {6, 7, 8} are 6, 7, and 8. Therefore:

X ∩ (Y ∩ Z) = {6, 7, 8}

Step 3: Calculate X ∩ Y

Next, we'll calculate the intersection of sets X and Y. Again, we're looking for elements present in both sets:

• X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• Y = {2, 4, 6, 7, 8}

The elements that appear in both X and Y are 2, 4, 6, 7, and 8. So:

X ∩ Y = {2, 4, 6, 7, 8}

Step 4: Calculate (X ∩ Y) ∩ Z

Now, we need to find the intersection of (X ∩ Y) with Z. We have:

• X ∩ Y = {2, 4, 6, 7, 8}
• Z = {5, 6, 7, 8}

The elements common to both {2, 4, 6, 7, 8} and Z are 6, 7, and 8. Thus:

(X ∩ Y) ∩ Z = {6, 7, 8}

Step 5: Compare the Results

Finally, let's compare the results we obtained:

• X ∩ (Y ∩ Z) = {6, 7, 8}
• (X ∩ Y) ∩ Z = {6, 7, 8}

As you can see, both sides of the equation are equal. This means we have successfully verified the associative law of intersection for sets X, Y, and Z!

Conclusion

Great job, guys! We've walked through the process of verifying the associative law of intersection for our sets X, Y, and Z. By systematically calculating each side of the equation and comparing the results, we've confirmed that the law holds true. Set theory can seem a bit abstract at first, but with practice, you'll become more comfortable with these concepts. Keep up the great work, and I'll see you in the next exploration!