# Question The Cartesian product $A \times B$ is defined as the set of all ordered pairs (a,b) where $a \in A$ and $b \in B$. Explain why the concept of "ordered" is critical to the definition and how this distinguishes Cartesian products from other set operations. ## Answer Choices (A) Ordering ensures that $|A \times B| = |A| \cdot |B|$ rather than $|A| + |B|$ (B) Ordering makes $(a,b) \neq (b,a)$ when $a \neq b$, creating a structure that can represent relations and functions, unlike unordered operations (C) Ordering guarantees that the Cartesian product is associative (D) Ordering prevents duplicate elements in the product set (E) Ordering allows the Cartesian product to be commutative like union and intersection # Solution The critical aspect of "ordered" in ordered pairs is that $(a,b)$ and $(b,a)$ are considered distinct elements unless $a = b$. This is fundamentally different from unordered pairs or sets where $\{a,b\} = \{b,a\}$. This ordering is essential because: 1. **Enables representation of relations**: A relation from A to B is defined as a subset of $A \times B$. The ordered nature allows us to distinguish "a is related to b" from "b is related to a," which is crucial for modeling directional relationships. 2. **Provides foundation for functions**: A function $f: A \to B$ is a special relation where each element of A appears exactly once as the first component of an ordered pair. Without ordering, we couldn't distinguish input from output. 3. **Creates asymmetric structure**: Unlike union ($A \cup B = B \cup A$) or intersection ($A \cap B = B \cap A$), the Cartesian product is generally not commutative: $A \times B \neq B \times A$ (unless A = B). This asymmetry reflects the directional nature of many mathematical structures. The other options are either incorrect (the product isn't commutative, ordering doesn't directly relate to preventing duplicates) or confuse consequence with purpose (while cardinality multiplication is true, it's not why ordering is critical). **Answer: (B)**