1. What an Ordered Pair Means
An ordered pair is a pair of two elements written in a fixed order. The first element comes before the second, and changing their order forms a completely different pair.
Ordered pairs are written like:
\( (a, b) \)
2. Why Order Matters
The important thing about ordered pairs is that the positions of the elements are not interchangeable. The first element has a different role from the second element. So, even if the same values are used, swapping them creates a new pair.
2.1. Examples
- \( (2,5) \neq (5,2) \)
- \( (a,b) \neq (b,a) \)
- Only when both elements are equal do we get the same pair: \( (3,3) = (3,3) \)
3. First and Second Elements
In the pair \( (a,b) \):
- a is the first element
- b is the second element
This structure helps define directions, connections, or mappings.
4. When Two Ordered Pairs Are Equal
Two ordered pairs are equal only if both their corresponding elements match exactly.
4.1. Condition
\( (a,b) = (c,d) \iff a = c \text{ and } b = d \)
4.2. Examples
- \( (4,9) = (4,9) \)
- \( (4,9) \neq (9,4) \)
5. Ordered Pairs in Coordinates
Ordered pairs are used to mark points on a plane. A pair like \( (x,y) \) shows position, where x is the horizontal value and y is the vertical value.
5.1. Example
(3,5) \text{ means x = 3, y = 5}
6. Ordered Pairs in Relations
Relations use ordered pairs to show how elements are connected. Each pair represents one link between two elements. This makes ordered pairs the basic building blocks of relations.
6.1. Example
R = \{(1,2), (2,3), (3,4)\}
This tells us that 1 is related to 2, 2 is related to 3, and 3 is related to 4.
7. Important Notes
Some simple points to remember:
7.1. Key Points
- Order cannot be changed.
- Equality depends on matching both positions.
- Used in coordinates, mappings, and relations.
- Forms the base of the Cartesian product.