Sets A & B: Define Correspondence & Build Graph

Let's dive into a fun math problem involving sets, relations, and graphs! We've got two sets, A and B, and a specific rule that connects them. Our mission is to figure out which elements from these sets are related according to that rule, visualize this relationship on a graph, and then list out all the pairs that make the rule true. Ready? Let's get started!

Understanding the Sets and the Relation

First, let's clearly define our sets. We have:

• A = {3, 6, 9, 12}
• B = {1, 4, 7}

And the relation Q is: "a is less than b by 2", where 'a' belongs to set A and 'b' belongs to set B. In mathematical terms, we can write this as:

• a ∈ A, b ∈ B, and a = b - 2

Now that we have a solid grasp of what we're working with, let's move on to defining the correspondence.

Defining the Correspondence

The correspondence between sets A and B based on relation Q means we need to find all pairs (a, b) where 'a' is an element of A, 'b' is an element of B, and the condition a = b - 2 holds true. Essentially, we're looking for pairs where the element from set A is 2 less than the element from set B. Let's test each element of A against each element of B to see which pairs satisfy this condition.

• If a = 3: We need to find a 'b' in B such that 3 = b - 2, which means b = 5. However, 5 is not in set B, so there's no correspondence here.
• If a = 6: We need to find a 'b' in B such that 6 = b - 2, which means b = 8. Again, 8 is not in set B, so no correspondence here either.
• If a = 9: We need to find a 'b' in B such that 9 = b - 2, which means b = 11. And yet again, 11 is not in set B, so there's no correspondence.
• If a = 12: We need to find a 'b' in B such that 12 = b - 2, which means b = 14. Still no luck, 14 isn't in set B.

It seems like our initial approach didn't yield any direct matches within the given sets. Let's re-examine the relation Q: "a is less than b by 2". This actually means b = a + 2. We should've been looking for elements in B that are 2 more than elements in A. My bad, guys!

Let's try this again with the correct interpretation:

• If a = 3: We need to find a 'b' in B such that b = 3 + 2, which means b = 5. Still no correspondence because 5 isn't in B.
• If a = 6: We need to find a 'b' in B such that b = 6 + 2, which means b = 8. No correspondence here either, as 8 isn't in B.
• If a = 9: We need to find a 'b' in B such that b = 9 + 2, which means b = 11. Still no luck; 11 is not a member of B.
• If a = 12: We need to find a 'b' in B such that b = 12 + 2, which means b = 14. Sadly, 14 is not within set B.

Okay, it looks like we're still striking out. Perhaps there was a typo in the original problem, or maybe the sets were intentionally designed to have no direct correspondences based on the given relation. Let's hold onto this result as it stands for now. Now, let's construct a graph based on our current (lack of) findings. While the math may not fully add up given these specific numbers, the methodology remains insightful.

Constructing the Graph of the Correspondence

A graph of a correspondence visually represents the relationship between elements of two sets. In this case, we'll represent the elements of set A on one axis and the elements of set B on the other axis. If a pair (a, b) satisfies the relation Q, we'll mark that point on the graph. Since we didn't find any pairs that actually satisfy the relation Q with the given sets, our graph will essentially be empty, illustrating that there's no direct correspondence between the elements of A and B based on the condition "a is less than b by 2" (or b is 'a' plus 2).

Here's how you'd generally construct such a graph, even though, in our case, it will be mostly illustrative:

1. Draw the Axes: Draw two axes. Label one axis 'A' and the other 'B'.
2. Mark the Elements: Mark the elements of set A (3, 6, 9, 12) along the 'A' axis and the elements of set B (1, 4, 7) along the 'B' axis.
3. Plot the Pairs: For each pair (a, b) that satisfies the relation Q (which, in our case, should have been a = b - 2 or b = a + 2), plot a point at the coordinates (a, b) on the graph.
4. No Points (in our case): Since none of our pairs matched up, you wouldn't plot any points on the graph. The graph would simply show the axes with the elements of A and B marked, but no connections between them.

If we had found matching pairs, the graph would show those points, giving us a visual representation of the correspondence.

Listing the Pairs of Elements

Finally, let's list all the pairs of elements from sets A and B that are in the correspondence. Based on our analysis, and given the original sets and relation, there are no pairs (a, b) that satisfy the condition "a is less than b by 2" (or b = a + 2). Therefore, the list of pairs is empty.

Correspondence Pairs: None

So, to recap, while we diligently tried to find matching pairs, construct a graph, and list the corresponding elements, the initial parameters of the problem (the specific elements within sets A and B, and the relation "a is less than b by 2") resulted in no direct correspondences. The approach and methodology were sound. Hopefully, this explanation gave you some insight into a complex subject!

If the sets or the relation were different, we would have found matching pairs, created a graph with plotted points, and listed the corresponding elements accordingly. The important aspect is understanding the process and how to apply it to different scenarios.