91Ó°ÊÓ

Imagine you have two sets, A and B. The union of sets, denoted as \( A \cup B \), is a new set that contains all elements from both A and B. This means any element that appears in at least one of the sets will be in the union. It's like combining two groups of friends into one large group.

In the given exercise, set A is defined as all points \((x, y)\) where \( 0 < x < 3 \) and \( 0 < y < 3 \), forming a square in the coordinate plane. Set B consists of all points \((x, y)\) where \( 2 < x < 4 \) and \( 2 < y < 4 \), which forms another square. The union \( A \cup B \) will include all the points that lie in either or both of these squares.

When you visualize this, you'll see that the union spans a square that goes from the smallest x and y value in A (0, 0) to the largest x and y value in B (4, 4). Thus, \( A \cup B \) covers a larger area, combining both initial squares into one complete region.

For two sets A and B, their intersection denoted as \( A \cap B \), includes only those elements that are present in both sets. Think of it as the shared space between two overlapping circles.

In this specific exercise, the intersection \( A \cap B \) is the set of points that lie within both set A – \( 0 < x < 3 \), \( 0 < y < 3 \) – and set B – \( 2 < x < 4 \), \( 2 < y < 4 \). The overlapping part is crucial here. Both sets overlap in the region where \( 2 < x < 3 \) and \( 2 < y < 3 \).

• This region forms a smaller square inside the coordinate plane.
• This new set is where both conditions for A and B are satisfied simultaneously.

Visualizing this, you notice that \( A \cap B \) creates a smaller square, showcasing the "common ground" shared by A and B.

Coordinate geometry is a branch of mathematics where geometric figures are represented using a coordinate system. This often involves locating points, lines, and curves by using ordered pairs \((x, y)\) on a plane, which are like directions on a map.

In this particular exercise, you're using coordinate geometry to define the regions of sets A and B. Each set represents a different rectangle on the plane, defined by specific x and y boundaries.

• A's rectangle stretches from x = 0 to x = 3 and y = 0 to y = 3.
• B's rectangle moves from x = 2 to x = 4 and y = 2 to y = 4.

Coordinate geometry helps you easily translate these algebraic descriptions (inequalities involving x and y) into tangible shapes on the xy-plane. By sketching these shapes, one can visually explore relationships like unions and intersections, making abstract ideas more concrete and approachable.