Cartesian product of two set is equivalence relation?

The Cartesian product of two sets is not necessarily an equivalence relation. An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity. However, the Cartesian product of two sets \( A \) and \( B \) (denoted as \( A \times B \)) simply consists of all ordered pairs \((a, b)\) where \( a \in A \) and \( b \in B \), and does not inherently satisfy these properties.

### Cartesian Product
Given two sets \( A \) and \( B \):
\[ A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\} \]

### Example
Let \( A = \{1, 2\} \) and \( B = \{3, 4\} \). The Cartesian product \( A \times B \) is:
\[ A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\} \]

### Equivalence Relation Properties
To check if \( A \times B \) is an equivalence relation, it must satisfy:

1. **Reflexivity**: Each element must be related to itself.
   - For \( (a, b) \) to be reflexive on \( A \times B \), it would need pairs \((a, a)\) and \((b, b)\) to exist for all elements in \( A \) and \( B \).
   - However, the Cartesian product does not include pairs like \((1, 1)\) or \((3, 3)\) unless \( A = B \), and even then it must include all such pairs.

2. **Symmetry**: If \( (a, b) \in R \), then \( (b, a) \in R \).
   - The Cartesian product \( A \times B \) does not guarantee that if \( (1, 3) \in A \times B \), then \( (3, 1) \in A \times B \).

3. **Transitivity**: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
   - The Cartesian product does not ensure this condition. For example, if \( (1, 3) \in A \times B \) and \( (3, 4) \in A \times B \), there is no element \( 3 \) in \( A \) to consider the next pair, so transitivity isn't inherently satisfied.

### Conclusion
The Cartesian product of two sets \( A \) and \( B \), \( A \times B \), is not necessarily an equivalence relation. It is simply a set of ordered pairs and does not guarantee reflexivity, symmetry, or transitivity. An equivalence relation needs to be defined explicitly on a set to satisfy these properties.

Do you have any further questions or need more details? Here are some related questions you might consider:

1. What is an equivalence relation, and how is it defined?
2. How to construct an equivalence relation from a given set?
3. Can you provide examples of equivalence relations in mathematics?
4. How does the Cartesian product differ from an equivalence relation?
5. What are the applications of equivalence relations in real-world scenarios?

**Tip:** When dealing with relations, always verify the specific properties (reflexivity, symmetry, transitivity) to determine the nature of the relation rather than relying on general set operations like the Cartesian product.

Explore the relationship between Cartesian product and equivalence relation in set theory. Learn why Cartesian product of two sets does not constitute an equivalence relation, covering reflexivity, symmetry, and transitivity. Ideal for college-level students and enthusiasts of mathematical set theory.

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