Discussion Overview
The discussion revolves around the properties of two binary relations, R and S. R is defined on the natural numbers (N) with the condition that aRb if and only if a <= b and b <= a + 5. S is defined on the integers (Z) and aSb if and only if a - b is an even integer. Participants explore whether these relations are reflexive, symmetric, antisymmetric, and transitive.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants question the reflexivity of R, suggesting that since a + 5 is not equal to a, it may not be reflexive.
- Others clarify that reflexivity means aRa must hold for all a in N, and argue that R is reflexive since a <= a and a <= a + 5 are true for any natural number a.
- There is confusion regarding the symmetry of R, with some noting that if a < b, then it does not imply b < a, raising questions about the relation's properties.
- Participants discuss the antisymmetric property, questioning whether the example of a = 4 and b = 5 violates antisymmetry, while others clarify that if 4R5 and 5R4 do not hold, it does not violate antisymmetry.
- Concerns are raised about the transitivity of R, with one participant suggesting that since 1R5 and 5R11 do not imply 1R11, R may not be transitive.
- One participant suggests testing specific pairs from a limited set of natural numbers to better understand the relation R.
Areas of Agreement / Disagreement
Participants express differing views on the properties of R, particularly regarding reflexivity, symmetry, antisymmetry, and transitivity. There is no consensus on these properties, and the discussion remains unresolved.
Contextual Notes
Some participants express uncertainty about the definitions and implications of reflexivity, symmetry, antisymmetry, and transitivity, indicating a need for further clarification and exploration of these concepts.