Set Relations: Arrow, Cartesian, And Ordered Pairs

by Tim Redaksi 51 views
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Let's dive into a fun math problem involving sets and relations! We've got set A with numbers from 10 to 15, and set B with numbers from 4 to 9. Our mission is to show how the relation "greater than" connects these two sets using different methods: arrow diagrams, Cartesian diagrams, and ordered pairs.

Understanding the Problem

Hey guys, before we jump into the solutions, let's make sure we understand what's being asked. We have two sets of numbers:

  • Set A = {10, 11, 12, 13, 14, 15}
  • Set B = {4, 5, 6, 7, 8, 9}

The relation between these sets is "greater than." This means we need to find all the pairs of numbers (a, b) where 'a' is from set A, 'b' is from set B, and 'a' is greater than 'b'. We'll then represent these pairs using different visual and symbolic methods.

a. Arrow Diagram

An arrow diagram is a visual way to represent the relation between two sets. We draw each set as a separate shape (usually an oval or rectangle) and then draw arrows from elements in set A to elements in set B if the relation "greater than" holds.

Constructing the Arrow Diagram

  1. Draw two ovals (or rectangles). Label one as "A" and the other as "B".

  2. Write the elements of each set inside their respective ovals. So, in oval A, we'll have 10, 11, 12, 13, 14, and 15. In oval B, we'll have 4, 5, 6, 7, 8, and 9.

  3. Draw arrows. This is the crucial part. For each number in set A, we need to check which numbers in set B are smaller than it. If a number in B is smaller, we draw an arrow from the number in A to that number in B.

    • From 10, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 10 is greater than all of these).
    • From 11, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 11 is greater than all of these).
    • From 12, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 12 is greater than all of these).
    • From 13, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 13 is greater than all of these).
    • From 14, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 14 is greater than all of these).
    • From 15, we draw arrows to 4, 5, 6, 7, 8, and 9 (since 15 is greater than all of these).

Key Points for the Arrow Diagram

  • Each arrow represents one instance of the "greater than" relation.
  • If there's no arrow between two numbers, it means the relation doesn't hold (e.g., there's no arrow from 4 in set B to any number in set A because no number in set A is smaller than 4).
  • The arrow diagram provides a clear visual representation of which elements in set A are related to which elements in set B based on the given relation.

The arrow diagram is complete, showcasing all instances where an element from set A is greater than an element from set B. This visual representation helps to quickly grasp the relationship between the two sets.

b. Cartesian Diagram

A Cartesian diagram, also known as a coordinate diagram, is another way to represent the relation between two sets. Instead of arrows, we use points on a graph.

Constructing the Cartesian Diagram

  1. Draw a coordinate plane. Label the x-axis as "B" and the y-axis as "A". This means the horizontal axis will represent the elements of set B, and the vertical axis will represent the elements of set A.

  2. Mark the elements of each set on their respective axes. On the x-axis (B), mark points for 4, 5, 6, 7, 8, and 9. On the y-axis (A), mark points for 10, 11, 12, 13, 14, and 15.

  3. Plot the points. For each pair (a, b) where 'a' is from set A, 'b' is from set B, and 'a' > 'b', plot a point at the coordinates (b, a). So, for example:

    • Since 10 > 4, plot a point at (4, 10).

    • Since 10 > 5, plot a point at (5, 10).

    • Since 10 > 6, plot a point at (6, 10).

    • Since 10 > 7, plot a point at (7, 10).

    • Since 10 > 8, plot a point at (8, 10).

    • Since 10 > 9, plot a point at (9, 10).

    • Repeat this process for 11, 12, 13, 14, and 15, plotting all points (b, a) where a > b.

Key Points for the Cartesian Diagram

  • Each point on the graph represents one instance of the "greater than" relation.
  • The x-coordinate of each point comes from set B, and the y-coordinate comes from set A.
  • The Cartesian diagram provides a visual representation of the relation as a set of points on a coordinate plane.

The Cartesian diagram is now complete, showing all the points that satisfy the condition where an element from set A is greater than an element from set B. It provides another way to visualize the relationship between the two sets.

c. Set of Ordered Pairs

The set of ordered pairs is a symbolic way to represent the relation between two sets. We simply list all the pairs (a, b) where 'a' is from set A, 'b' is from set B, and 'a' is greater than 'b'.

Constructing the Set of Ordered Pairs

We need to identify all pairs (a, b) that satisfy the condition a > b. Let's list them:

  • (10, 4), (10, 5), (10, 6), (10, 7), (10, 8), (10, 9)
  • (11, 4), (11, 5), (11, 6), (11, 7), (11, 8), (11, 9)
  • (12, 4), (12, 5), (12, 6), (12, 7), (12, 8), (12, 9)
  • (13, 4), (13, 5), (13, 6), (13, 7), (13, 8), (13, 9)
  • (14, 4), (14, 5), (14, 6), (14, 7), (14, 8), (14, 9)
  • (15, 4), (15, 5), (15, 6), (15, 7), (15, 8), (15, 9)

Now, we write these pairs as a set:

{(10, 4), (10, 5), (10, 6), (10, 7), (10, 8), (10, 9), (11, 4), (11, 5), (11, 6), (11, 7), (11, 8), (11, 9), (12, 4), (12, 5), (12, 6), (12, 7), (12, 8), (12, 9), (13, 4), (13, 5), (13, 6), (13, 7), (13, 8), (13, 9), (14, 4), (14, 5), (14, 6), (14, 7), (14, 8), (14, 9), (15, 4), (15, 5), (15, 6), (15, 7), (15, 8), (15, 9)}

Key Points for the Set of Ordered Pairs

  • Each ordered pair (a, b) indicates that 'a' is related to 'b' according to the specified relation (in this case,