QuestionICS 6B homework submissions are timestamped on submission up to an accuracy of 0.01 s . Consider the domain of all homework submissions over a relation where if is submitted before . It is possible that close to a deadline with an increase in the number of submissions per unit time the server notes multiple submissions to have been submitted on the same time. Check all that apply to and you ONLY need to explain which order it is (no need to justify for the other relation properties).
Symmetric Transitive
Reflexive
Anti-symmetric Not transitive
Anti-reflexive
Neither Symmetric
nor
Anti-reflexive
Anti-symmetric
Partial Order Strict Order Total Order
Explanation:
Studdy Solution
STEP 1
1. The relation is defined on the domain of all homework submissions.
2. means that submission is submitted before submission .
3. Submissions can be timestamped to an accuracy of 0.01 seconds.
4. Multiple submissions can have the same timestamp.
STEP 2
1. Determine if the relation is reflexive or anti-reflexive.
2. Determine if the relation is symmetric or anti-symmetric.
3. Determine if the relation is transitive.
4. Classify the relation as a partial order, strict order, or total order.
STEP 3
Determine if the relation is reflexive or anti-reflexive.
- Reflexive: For all , must hold. In this context, a submission cannot be submitted before itself, so is false for all .
- Anti-reflexive: For all , must not hold. Since a submission cannot be submitted before itself, this condition is satisfied.
Conclusion: The relation is anti-reflexive.
STEP 4
Determine if the relation is symmetric or anti-symmetric.
- Symmetric: If , then must hold. If is submitted before , then it cannot be true that is submitted before , so is not symmetric.
- Anti-symmetric: If and , then must hold. Since implies is submitted before , and implies is submitted before , both cannot be true unless , which is not possible due to anti-reflexivity.
Conclusion: The relation is anti-symmetric.
STEP 5
Determine if the relation is transitive.
- Transitive: If and , then must hold. If is submitted before and is submitted before , then is submitted before , satisfying transitivity.
Conclusion: The relation is transitive.
STEP 6
Classify the relation as a partial order, strict order, or total order.
- Partial Order: A relation that is reflexive, anti-symmetric, and transitive. Since is anti-reflexive, it cannot be a partial order.
- Strict Order: A relation that is anti-reflexive, anti-symmetric, and transitive. satisfies all these properties.
- Total Order: A strict order where for any and , either , , or must hold. Since multiple submissions can have the same timestamp, is not a total order.
Conclusion: The relation is a strict order.
The relation is a strict order.
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