Here is a clear, simple, and exam-focused explanation of Ordered Pairs in Sets.

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⭐ Ordered Pairs in Sets

An ordered pair is a pair of elements written in a specific order, usually as:

[
(a, b)
]

It consists of two elements:

• a = first element
• b = second element

The order matters.
This means:

[
(a, b) \neq (b, a) \quad \text{in general}
]

unless a = b.

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📌 Example

[
(2, 5) \neq (5, 2)
]

because the first element and second element are different.

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⭐ Why Ordered Pairs?

Ordered pairs are used in:

✔ Relations
✔ Functions
✔ Cartesian products
✔ Coordinate geometry
✔ Database tuples
✔ Graph theory (edges as ordered pairs in directed graphs)

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⭐ Equality of Ordered Pairs

Two ordered pairs are equal if and only if:

[
(a, b) = (c, d) \quad \text{when } a = c \text{ and } b = d
]

Example:

[
(3, 7) = (3, 7) \quad \text{(equal)}
]

[
(3, 7) \neq (7, 3) \quad \text{(not equal)}
]

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⭐ Ordered Pairs vs Unordered Pairs

Ordered Pair:

[
(a, b) \neq (b, a)
]

Unordered Pair (Set):

[
{a, b} = {b, a}
]

Sets ignore order; ordered pairs do not.

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⭐ Cartesian Product Creates Ordered Pairs

If
[
A = {1,2}, \quad B = {x,y}
]

then
[
A \times B = {(1,x), (1,y), (2,x), (2,y)}
]

Each pair is an ordered pair (first from A, second from B).

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⭐ Applications in Discrete Structures

1. Relations
   A relation is a set of ordered pairs.
   Example: R = {(1,2), (2,3)}
2. Functions
   A function is a special relation where each x has one y.
3. Directed Graphs
   Edges are ordered pairs (u, v).
4. Coordinates
   (x, y) represent points on the plane.

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⭐ Why Order is Important

Example:

• (student, course)
  ≠
• (course, student)

Because their meaning changes completely.

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