- #1
knowLittle
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Homework Statement
Is this relation, R, on ## S= \{ 1, 2, 3 \} \\ R = \{ (1,1), (2,2) , (3,3) \}##
Symmetric?
It is obvious that it is reflexive.
knowLittle said:Homework Statement
Is this relation, R, on ## S= \{ 1, 2, 3 \} \\ R = \{ (1,1), (2,2) , (3,3) \}##
Symmetric?
It is obvious that it is reflexive.
knowLittle said:Wait, so my R is an equivalence relation then? Supposedly, it partitions the set into disjoint classes. I guess that my classes would be [1], [2], [3]?
Reflexivity is a concept in mathematics and logic that refers to the property of a relation or operation being applied to itself. In other words, an element is related to itself. Symmetry, on the other hand, refers to a property of a relation or operation where it remains unchanged when the elements are swapped. Reflexivity implies symmetry because if every element is related to itself, then swapping the elements will not change the relation.
One example is the equality relation in mathematics. If we consider the relation "equal to" and apply it to any element, say 5, then it is reflexive because 5 is equal to itself. Now, if we swap the elements and compare 5 to another number, say 7, the relation is still true because 5 and 7 are not equal. Thus, reflexivity implies symmetry in this case.
Reflexivity and symmetry are commonly used in the fields of mathematics, computer science, and physics. In mathematics, these concepts are important in the study of abstract algebra, graph theory, and topology. In computer science, they are used in the development of algorithms and data structures. In physics, they are applied in the study of symmetry in physical laws and principles.
No, reflexivity is not always a necessary condition for symmetry. There are cases where a relation can be symmetric without being reflexive. For example, the relation "greater than" is symmetric, but it is not reflexive because a number cannot be greater than itself.
Reflexivity implies transitivity because if a relation is reflexive, then every element is related to itself. And if a relation is transitive, then if two elements are related and the second element is related to a third element, then the first element is also related to the third element. This can be seen as a chain of reflexivity, where each element is related to itself and to the next element, ultimately leading to transitivity.