SUMMARY
The discussion centers on determining whether the relation defined by \( aRb \) if \( ab \geq 0 \) is an equivalence relation on the set of integers \( S \). The three properties required for equivalence relations—reflexivity, symmetry, and transitivity—are examined. Reflexivity is satisfied since \( aRa \) holds true for all integers \( a \) (as \( a \cdot a \geq 0 \)). Symmetry is also confirmed, as \( aRb \) implies \( bRa \) due to the commutative property of multiplication. Lastly, transitivity is established since if \( aRb \) and \( bRc \), then \( aRc \) follows from the non-negativity of the products.
PREREQUISITES
- Understanding of equivalence relations in mathematics
- Basic knowledge of integer properties
- Familiarity with reflexivity, symmetry, and transitivity concepts
- Ability to manipulate and analyze inequalities
NEXT STEPS
- Study the properties of equivalence relations in more depth
- Explore examples of equivalence relations on different sets
- Learn about the implications of equivalence classes
- Investigate other mathematical relations and their classifications
USEFUL FOR
Students studying abstract algebra, mathematics educators, and anyone interested in understanding equivalence relations and their applications in set theory.