Math Problem Statement
Give an example of relation on {1,2,3,4} that is not reflexive, symmetric, not antisymmetric, and transitive.
Solution
To construct a relation on the set that is not reflexive, not symmetric, not antisymmetric, and not transitive, we need to define the relation carefully to violate all of these properties.
Let's define the relation on the set as follows:
Now, let's check the properties:
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Not Reflexive: A relation is reflexive if for every element , is in the relation. However, in this case, there are no pairs of the form , , , or in . Hence, the relation is not reflexive.
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Not Symmetric: A relation is symmetric if whenever is in the relation, must also be in the relation. In , we have but is in the relation, which makes the relation appear symmetric. However, the pair is in the relation, but is not. So, the relation is not symmetric.
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Not Antisymmetric: A relation is antisymmetric if for any pair of elements and in the relation, . In our relation, we have both and in , but . Thus, the relation is not antisymmetric.
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Not Transitive: A relation is transitive if whenever and are in the relation, must also be in the relation. In this case, and are in , but is not. Hence, the relation is not transitive.
Summary:
The relation on the set is not reflexive, not symmetric, not antisymmetric, and not transitive.
Would you like more details on any of these properties or further examples?
Here are some follow-up questions to explore:
- What are the conditions for a relation to be reflexive, symmetric, antisymmetric, and transitive?
- Can you provide an example of a relation that is both symmetric and transitive but not reflexive?
- How would you construct a relation that is reflexive but neither symmetric nor transitive?
- What are the differences between antisymmetry and symmetry in relations?
- How do you check if a relation on a set is transitive?
Tip: To easily check for transitivity, try finding "chains" in the relation where and are present, and then verify whether is also in the relation.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Relations
Properties of Relations
Formulas
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Theorems
Reflexivity
Symmetry
Antisymmetry
Transitivity
Suitable Grade Level
Grades 9-12
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