Quick Revision Notes — Ordered Pairs & Cartesian Products
60-Second Summary
The core takeaways in under a minute
An ordered pair $(a, b)$ is order-sensitive, representing coordinates in Cartesian geometry; it is equal to $(c, d)$ if and only if $a = c$ and $b = d$. The Cartesian Product $A \times B$ is the set of all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$. Cardinality follows $n(A \times B) = n(A) \times n(B)$, and the number of subsets is $2^{n(A \times B)}$. Cartesian products distribute over union ($\cup$) and intersection ($\cap$). It is not commutative: $A \times B \neq B \times A$ in general.
Ordered Pair vs. Set Comparison
Key differences in notation and properties
| Feature | Ordered Pair $(a, b)$ | Set $\{a, b\}$ |
|---|---|---|
| Order Sensitivity | Sensitive: $(a, b) \neq (b, a)$ (if $a \neq b$) | Insensitive: $\{a, b\} = \{b, a\}$ |
| Equal Elements | $(a, a)$ is a valid pair | $\{a, a\} = \{a\}$ (duplicates collapse) |
| Notation | Parenthesis: $(\text{ })$ | Curly brackets: $\{\text{ }\}$ |
Cartesian Product Properties Matrix
Identities to memorize
| Identity / Property | Mathematical Formulation |
|---|---|
| Cardinality | $n(A \times B) = n(A) \times n(B)$ |
| Union Distributivity | $A \times (B \cup C) = (A \times B) \cup (A \times C)$ |
| Intersection Distributivity | $A \times (B \cap C) = (A \times B) \cap (A \times C)$ |
| Total Subsets | Number of subsets $= 2^{n(A) \times n(B)}$ |
| Empty Set Product | $A \times \emptyset = \emptyset \times B = \emptyset$ |
Common Mistakes to Avoid
Watch out for these classic exam traps!
Assuming $A \times B = B \times A$. The elements of $A \times B$ are $(a, b)$, while elements of $B \times A$ are $(b, a)$. Since order matters, the sets are different.
Writing a Cartesian product element as $\{a, b\}$. Elements of Cartesian products are always written as ordered pairs with parentheses: $(a, b)$.
Confusing the number of elements with the number of subsets. $n(A \times B)$ is the number of elements. The number of subsets is $2^{n(A \times B)}$.