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.)
This is what I have so far, can you check to see if this is correct?
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, xRx 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.
Your explanation is partially correct, but there are some mistakes.
To prove the statement "R is an equivalence relation if and only if R is both reflexive and orbital," we need to show two different 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 break down these two implications:
1. If R is an equivalence relation, then R is both reflexive and orbital:
To prove this, you need to explain how the properties of an equivalence relation (reflexivity, symmetry, and transitivity) imply that R is both reflexive and orbital. You correctly mentioned that an equivalence relation is reflexive, symmetric, and transitive, but you didn't explicitly explain why symmetry and transitivity imply that R is orbital. Here's how you can elaborate on that:
- Symmetry: If R is symmetric, i.e., for every x and y, if xRy, then yRx, we can see that if xRy and yRz, then applying symmetry, we get yRx. Since we now have yRz and yRx, the transitivity property of an equivalence relation implies that zRx holds. Therefore, R is orbital.
2. If R is both reflexive and orbital, then R is an equivalence relation:
To prove this, you need to explain how the properties of being reflexive and orbital directly imply the properties of an equivalence relation (reflexivity, symmetry, and transitivity). You mentioned reflexivity and orbital property, but you didn't explain how the orbital property guarantees symmetry and transitivity. Here's how you can elaborate on that:
- Reflexivity: If xRx holds for all elements x, we directly see that R is reflexive.
- Orbital: If xRy and yRx hold, we can use the orbital property to conclude that zRx must hold. This implies transitivity, as for any x, y, and z, if xRy and yRx, then zRx.
So, combining reflexivity and the orbital property, we see that R satisfies the properties of an equivalence relation: reflexivity, symmetry (from orbital), and transitivity (from orbital).
Taking both implications together, we can conclude that R is an equivalence relation if and only if R is both reflexive and orbital.