Call a relation R “orbital” if x R yand y R zimply z R x. 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 biconditional. So there are actually two different implications to show here.)
This is what I did, but I don't think that it is the right approach:
If R is an equivalence relation then R has to be reflexive, symmetric, and transitive. This means that xRx which is reflexive. Also, xRy and yRx which means that it is symmetric.Then, R is also orbital or transitive if xRy and yRz which implies that zRx. This means that R is an equivalence relation. If R is reflexive then xRx. If R is orbital, then xRy and yRz which implies that xRz. This means that R is an equivalence relation.
To prove that a relation R is an equivalence relation if and only if R is both reflexive and orbital, we need to demonstrate two implications:
1. If R is an equivalence relation, then R is both reflexive and orbital.
2. If R is both reflexive and orbital, then R is an equivalence relation.
Let's take these two implications one by one.
1. If R is an equivalence relation, then R is both reflexive and orbital:
To prove this, we need to show that R satisfies the properties of reflexivity and orbitality.
a) Reflexivity: If R is an equivalence relation, then it must be reflexive. This means that for every element x in the set, we have x R x.
b) Orbitality: If R is an equivalence relation, then it must be orbital. This means that if x R y and y R z, then z R x for any elements x, y, and z in the set.
Since R satisfies both reflexivity and orbitality, we can conclude that R is both reflexive and orbital if it is an equivalence relation.
2. If R is both reflexive and orbital, then R is an equivalence relation:
To prove this, we need to show that R satisfies the properties of reflexivity, symmetry, and transitivity.
a) Reflexivity: We are given that R is reflexive, which means that for every element x in the set, we have x R x.
b) Symmetry: To prove symmetry, we need to show that if x R y, then y R x for any elements x and y in the set. Since R is orbital, we know that if x R y, then y R x. Hence, R is symmetric.
c) Transitivity: To prove transitivity, we need to show that if x R y and y R z, then x R z for any elements x, y, and z in the set. Given that R is orbital, we know that if x R y and y R z, then z R x. Since we already have x R z, this implies transitivity.
As R satisfies reflexivity, symmetry, and transitivity, we can conclude that R is an equivalence relation if it is both reflexive and orbital.
Therefore, we have shown both implications, and thus, R is an equivalence relation if and only if R is both reflexive and orbital.