1. What Are Ordered Pairs?
An ordered pair is a pair of two elements written in a specific order. The first element comes before the second, and changing their order creates a different pair.
Ordered pairs are written using round brackets:
\( (a, b) \)
2. Order Matters
The most important point about ordered pairs is that the order is fixed. This means:
2.1. Example
- \( (2, 5) \neq (5, 2) \)
- \( (a, b) \neq (b, a) \) unless a = b
Swapping the elements makes a new pair.
3. First and Second Elements
In an ordered pair \( (a, b) \):
- a is called the first element
- b is called the second element
Each element has its own position and meaning.
4. Examples of Ordered Pairs
Some simple examples:
4.1. Numeric Examples
- \( (3, 7) \)
- \( (1, 1) \)
- \( (-2, 5) \)
4.2. Algebraic Examples
- \( (x, y) \)
- \( (a, b) \)
- \( (p+1, q-3) \)
5. When Ordered Pairs Are Equal
Two ordered pairs are equal only when both their first elements match and both their second elements match.
5.1. Condition
\( (a, b) = (c, d) \iff a = c \text{ and } b = d \)
5.2. Example
- \( (4, 9) = (4, 9) \)
- \( (4, 9) \neq (9, 4) \)
6. Ordered Pairs in Coordinate Geometry
Ordered pairs are used to locate points on the coordinate plane:
\( (x, y) \)
x shows horizontal position and y shows vertical position.
7. Ordered Pairs in Relations
In set theory, ordered pairs are used to show relationships between elements. A relation is often written as a set of ordered pairs, such as:
\( R = \{(1,2), (2,3), (3,4)\} \)
Each pair connects one element to another.
8. Important Points
A few things to remember:
8.1. Key Ideas
- Order matters in an ordered pair.
- Changing the order creates a new pair.
- Two ordered pairs are equal only if corresponding elements match.
- Widely used in coordinates, relations, and functions.