Ordered Pairs & Cartesian Product: Concepts

Section 3.1

Introduction to Relations & Functions

In everyday life, we often observe associations between two quantities:

  • The marks obtained by a student in an MCQ test depends on the number of correct answers.
  • The area of a circle depends on its radius.
  • A person's salary may depend on years of experience.
  • The distance travelled at uniform speed depends on time.

All such dependencies can be described mathematically using relations and functions.

What is a Function?

Suppose every student in a school is allotted exactly one house. This connection rule is a function because each student maps to precisely one house. If instead a student is paired with one or more sports teams, it is simply a relation.

Section 3.2

Ordered Pairs

An ordered pair is a pair of objects written in a specific order. It is denoted by $(a, b)$, where $a$ is the first element and $b$ is the second element. Since order matters, in general, $(a, b) \neq (b, a)$ unless $a = b$.

Equality of Ordered Pairs:
Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if their corresponding components are equal:
$$(a, b) = (c, d) \iff a = c and b = d$$
Coordinate Geometry Connection

An ordered pair $(x, y)$ represents the position of a point in the Cartesian plane, where $x$ is the abscissa (horizontal coordinate) and $y$ is the ordinate (vertical coordinate).

Ex 1
Example 1: If $(2x + 1, y - 3) = (9, 4)$, find $x$ and $y$.
Show Solution
Since the ordered pairs are equal, equate their abscissas and ordinates:
$$2x + 1 = 9 \Rightarrow 2x = 8 \Rightarrow \mathbf{x = 4}$$
$$y - 3 = 4 \Rightarrow \mathbf{y = 7}$$
Section 3.3

Cartesian Product of Sets

Given two non-empty sets $A$ and $B$, their Cartesian Product $A × B$ (read as '$A$ cross $B$') is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. $$A × B = \{(a, b) : a \in A and b \in B\}$$ If either $A = \emptyset$ or $B = \emptyset$, then $A × B = \emptyset$.

Properties of Cartesian Product:

1. Cardinality: $n(A × B) = n(A) × n(B)$.
2. Distributive Laws:
    - $A × (B \cup C) = (A × B) \cup (A × C)$
    - $A × (B \cap C) = (A × B) \cap (A × C)$
3. Subset Relation: If $A \subseteq B$, then $A × C \subseteq B × C$ for any set $C$.
Non-Commutative Warning

The Cartesian product is not commutative: $A × B \neq B × A$ unless $A = B$ or one of the sets is empty.

Ex 2
Example 2: Let $A = \{1, 2, 3\}$ and $B = \{4, 5\}$. Find $A × B$ and $B × A$. Are they equal?
Show Solution
$$A × B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}$$ $$B × A = \{(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)\}$$ Since $(1, 4) \in A × B$ but $(1, 4) \notin B × A$, we have $\mathbf{A × B \neq B × A}$.
Ex 3
Example 3: If $n(A) = 4$ and $n(B) = 3$, find how many subsets will $A × B$ have?
Show Solution
The cardinality is:
$$n(A × B) = n(A) × n(B) = 4 × 3 = 12.$$
The number of subsets of a set of size $N$ is $2^N$.
Therefore, number of subsets of $A × B$ is:
$$\mathbf{212 = 4096}.$$
Ex 4
Example 4: If $A = \{a, b\}$ and $B = \{1, 2, 3\}$, write $A × B$ and verify $n(A × B) = n(A) × n(B)$.
Show Solution
$$A × B = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$$ Count elements:
- $n(A × B) = 6$.
- $n(A) = 2$ and $n(B) = 3$.
Since $6 = 2 × 3$, we have verified $n(A × B) = n(A) × n(B)$.
Ex 5
Example 5: If $A = \{1, 3, 5\}$ and $B = \{4, 9\}$. Write the ordered pairs of $A × B$ whose first element is less than the second element. What about $B × A$ satisfying the same order relation? Are they equal?
Show Solution
For $A × B$: We want $(a, b) \in A × B$ such that $a < b$.
The pairs are: $\mathbf{\{(1, 4), (1, 9), (3, 4), (3, 9), (5, 9)\}}$.

For $B × A$: We want $(b, a) \in B × A$ such that $b < a$.
- $b=4$: $4 < 5 \Rightarrow (4, 5)$ is valid.
- $b=9$: no element in $A$ is greater than 9.
The pairs are: $\mathbf{\{(4, 5)\}}$.

Since the sets of ordered pairs are different, they are **not equal**.
Ex 6
Example 6: Let $A = \{1, 3\}$, $B = \{2, 3, 4\}$ and $C = \{4, 5\}$. Write the set $A × (B \cap C)$ and verify $A × (B \cap C) = (A × B) \cap (A × C)$.
Show Solution
LHS:
$$B \cap C = \{2, 3, 4\} \cap \{4, 5\} = \{4\}$$ $$A × (B \cap C) = \{1, 3\} × \{4\} = \mathbf{\{(1, 4), (3, 4)\}}$$.

RHS:
$$A × B = \{(1, 2), (1, 3), (1, 4), (3, 2), (3, 3), (3, 4)\}$$ $$A × C = \{(1, 4), (1, 5), (3, 4), (3, 5)\}$$ $$(A × B) \cap (A × C) = \mathbf{\{(1, 4), (3, 4)\}}$$
Since LHS = RHS, property is verified.