Finding The Intersection Of Sets A And B

by Felix Dubois 41 views

Hey guys! Today, we're diving into a fascinating problem involving sets and coordinate planes. We're given three sets: U, A, and B, and our mission is to find the ordered pair that satisfies the intersection of sets A and B. Let's break it down step by step and make sure we understand every detail.

Understanding the Sets

First, let's define our sets clearly:

  • U: The Universal Set

    U represents the universal set, which includes all ordered pairs on a coordinate plane. Think of it as the entire playground of points where x and y coordinates can take any real value. This is our backdrop, the space in which our other sets exist. Understanding the universal set is crucial because it sets the boundaries for our problem. It's like saying we're looking for something within a specific room, not the entire house. So, anything we find must exist within this set of ordered pairs. For example, (0, 0), (1, 2), (-3, 5), and even irrational coordinates like (π, √2) are all members of U. This set is infinitely large, covering every possible point on the Cartesian plane. Therefore, when we consider subsets, we know they must draw their elements from this vast collection. The universal set acts as the ultimate reference for where our solutions can be found, ensuring we stay within the defined scope of the problem. Remember, without a clear understanding of U, we might wander into solutions that don't fit the context. The coordinate plane, with its x and y axes stretching infinitely in all directions, is the visual representation of this set. It’s the canvas on which our mathematical relationships will be drawn and analyzed.

  • A: Ordered Pair Solutions to y = x

    Set A consists of all ordered pairs (x, y) that satisfy the equation y = x. This means that for any point in set A, the x-coordinate and the y-coordinate are equal. Graphically, this set represents a straight line that passes through the origin (0, 0) and has a slope of 1. Points like (1, 1), (2, 2), (-3, -3), and (0, 0) are all members of A because they fit the condition that y is equal to x. This set represents a very specific relationship between the x and y values; they are always identical. It’s a diagonal line cutting through the coordinate plane, a visual representation of perfect equality between the two coordinates. Understanding this set is key to solving the problem, as it narrows down the possible solutions to those points where the x and y values align. To visualize this, imagine drawing a line on the coordinate plane where every point has the same x and y value. That line represents set A. The simplicity of the equation y = x belies the vast number of points that belong to this set; it's an infinite collection, but one that's tightly constrained by this fundamental equality.

  • B: Ordered Pair Solutions to y = 2x

    Set B is defined as the set of all ordered pairs (x, y) that satisfy the equation y = 2x. In this case, the y-coordinate is twice the x-coordinate. This set also represents a straight line passing through the origin, but with a steeper slope of 2. Points such as (1, 2), (2, 4), (-1, -2), and (0, 0) are members of B because they fulfill the condition y = 2x. Compared to set A, this set represents a different relationship between x and y; here, y is always double the value of x. This means the line representing set B will rise more quickly as x increases, making it steeper than the line for set A. Identifying the points that satisfy this relationship is crucial for finding the intersection between sets A and B. The line representing set B is another infinite collection of points, but its path across the coordinate plane is dictated by this doubling relationship. Visualizing this line alongside the line for set A can help us see where they might intersect, which will ultimately lead us to our solution.

Finding the Intersection: A ∩ B

The symbol "∩" represents the intersection of two sets. A ∩ B means we're looking for the ordered pairs that are members of both set A and set B. In other words, we need to find the points (x, y) that satisfy both equations:

  • y = x
  • y = 2x

To find these points, we can use a few methods, but the most straightforward is substitution. Since both equations are equal to y, we can set them equal to each other:

x = 2x

Now, let's solve for x:

x - 2x = 0

-x = 0

x = 0

Great! We found that x = 0. Now, we can substitute this value back into either equation to find y. Let's use y = x:

y = 0

So, the ordered pair that satisfies both equations is (0, 0).

This is the point where the lines represented by y = x and y = 2x intersect on the coordinate plane. It's the one point that belongs to both sets, making it the solution to A ∩ B. To truly grasp why (0, 0) is the only solution, think about the graphical representation of these equations. Both are straight lines that pass through the origin. Since they have different slopes (1 and 2), they will only intersect at one point, which is the origin itself. Any other point would either lie on the line y = x or the line y = 2x, but not both. Thus, the intersection is a single, definitive point. The algebraic method of substitution perfectly complements this graphical understanding, providing a concrete way to identify the intersection point. This intersection represents a unique solution, a point where the relationships defined by both equations hold true simultaneously.

Verifying the Solution

To be absolutely sure, let's plug (0, 0) back into both equations:

  • For y = x: 0 = 0 (True)
  • For y = 2x: 0 = 2 * 0 (True)

Yep, it checks out! (0, 0) satisfies both equations and is therefore the solution to A ∩ B.

This verification step is crucial in mathematics. It's like double-checking your work to ensure you haven't made any mistakes along the way. By plugging the solution back into the original equations, we confirm that it indeed satisfies all the conditions of the problem. This process solidifies our understanding and provides a sense of certainty that our answer is correct. It's not just about finding an answer; it's about proving that the answer is valid within the given mathematical framework. In the context of this problem, verifying (0, 0) ensures that it truly lies on both lines represented by the equations y = x and y = 2x. This reinforces the concept of intersection as the set of points that satisfy multiple conditions simultaneously. The act of verification transforms a potential solution into a confirmed one, strengthening our confidence in the mathematical process and outcome.

Conclusion

So, guys, the ordered pair that satisfies A ∩ B is (0, 0). We found this by understanding the definitions of the sets, setting the equations equal to each other, solving for x and y, and verifying our solution. Math can be like a cool puzzle, right? Keep practicing, and you'll become a pro at solving these kinds of problems!

This problem underscores the importance of understanding set theory and coordinate geometry. By carefully defining the sets and their relationships, we were able to systematically find the intersection point. The use of algebraic techniques, like substitution, further facilitated the solution. Remember, the key to solving mathematical problems lies in breaking them down into manageable steps, understanding the underlying concepts, and verifying your results. This approach not only leads to correct answers but also builds a deeper appreciation for the elegance and logic of mathematics. So, keep exploring, keep questioning, and keep solving!