Call a relation R “orbital” if xRy and yRz imply zRx. Prove that R is an equivalence relation if and only R is both reflexive and orbital. (Note that this is an “if and only if” statement, which is bi-conditional. So there are actually two different implications to show here.)
How do I start to prove that this is a relation?
To prove that a relation R is an equivalence relation, we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity: To show that R is reflexive, we need to prove that for every element x, xRx holds. In other words, every element must be related to itself.
2. Symmetry: To show that R is symmetric, we need to prove that if xRy holds, then yRx also holds. This means that if x is related to y, then y must also be related to x.
3. Transitivity: To show that R is transitive, we need to prove that for any three elements x, y, and z, if xRy and yRz hold, then xRz also holds. This means that if x is related to y, and y is related to z, then x must be related to z.
Given that R is defined as "orbital", we also need to prove that if R is both reflexive and orbital, then it is an equivalence relation.
Let's begin with the first implication, which is to prove that if R is an equivalence relation, then R is both reflexive and orbital.
Suppose R is an equivalence relation:
1. Reflexivity: To show that R is reflexive, we need to prove that xRx holds for every element x. Since R is an equivalence relation, reflexivity is guaranteed. Thus, R is reflexive.
2. Orbital Property: To prove that R is orbital, we need to show that if xRy and yRz hold, then zRx also holds. Since R is an equivalence relation, it implies that if x is related to y and y is related to z, then x is also related to z. Therefore, R satisfies the orbital property.
Therefore, if R is an equivalence relation, it is both reflexive and orbital.
Now, for the second implication, we need to prove that if R is both reflexive and orbital, then R is an equivalence relation.
Suppose R is both reflexive and orbital:
1. Reflexivity: Given that R is reflexive, it satisfies the first condition of an equivalence relation.
2. Symmetry: To show that R is symmetric, we need to prove that if xRy holds, then yRx also holds. Since R is orbital, if xRy, then yRx holds.
3. Transitivity: To show that R is transitive, we need to prove that if xRy and yRz hold, then xRz also holds. Since R is orbital, if xRy and yRz, then zRx. Thus, R satisfies the transitive property.
Therefore, if R is both reflexive and orbital, it is an equivalence relation.
In conclusion, we have proven both implications: (1) if R is an equivalence relation, then R is both reflexive and orbital, and (2) if R is both reflexive and orbital, then R is an equivalence relation.