Ordered Pairs and Their Equality

Definition Of Ordered Pairs

• An ordered pair is a pair of elements where the order of the elements is significant.
• It is typically written in the form (a, b), where:
  • 'a' is the first element (the first coordinate).
  • 'b' is the second element (the second coordinate).
• The first element corresponds to the x-coordinate and the second element to the y-coordinate in a Cartesian plane.

Equality Criteria

• Two ordered pairs (a, b) and (c, d) are considered equal if:
  • The first elements are equal: a = c
  • The second elements are equal: b = d
• Therefore, equality can be summarized as:
  • (a, b) = (c, d) if and only if a = c and b = d.

Examples Of Ordered Pairs

1. Example 1:

   • Ordered Pair: (2, 3)
   • First element: 2
   • Second element: 3

2. Example 2:

   • Ordered Pair: (5, -1)
   • First element: 5
   • Second element: -1

3. Equality Example:

   • Ordered Pairs: (4, 7) and (4, 7)
     • These pairs are equal since both first and second elements are the same.
   • Ordered Pairs: (1, 2) and (2, 1)
     • These pairs are not equal because the order of elements is different.

4. Real-World Context:

   • Ordered pairs can represent coordinates on a graph, e.g., (3, 4) indicates a point located 3 units along the x-axis and 4 units along the y-axis.

Definition Of Ordered Pairs

• An ordered pair consists of two elements where the sequence is important.
• Typically expressed as (a, b) with:
  • 'a' representing the first element, also known as the first coordinate.
  • 'b' representing the second element, or the second coordinate.
• In a Cartesian plane, 'a' corresponds to the x-coordinate and 'b' to the y-coordinate.

Equality Criteria

• Two ordered pairs, (a, b) and (c, d), are equal under specific conditions:
  • The first elements must be equal (a = c).
  • The second elements must also be equal (b = d).
• Equality can be summarized as:
  • (a, b) = (c, d) if and only if both conditions are satisfied: a = c and b = d.

Examples Of Ordered Pairs

• Example 1:
  • Ordered Pair: (2, 3)
    • First element: 2
    • Second element: 3
• Example 2:
  • Ordered Pair: (5, -1)
    • First element: 5
    • Second element: -1
• Equality Example:
  • Ordered Pairs (4, 7) and (4, 7) are equal since both corresponding elements are identical.
  • Ordered Pairs (1, 2) and (2, 1) are not equal due to the different order of elements.
• Real-World Context:
  • Ordered pairs can denote coordinates on a graph; for instance, (3, 4) signifies a point that is 3 units on the x-axis and 4 units on the y-axis.