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?
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.
To prove that a relation is an equivalence relation, we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.
Let's start with the first implication: "If R is an orbital relation, then R is reflexive." This means we need to show that if R is orbital, then it must also be reflexive.
To begin the proof, we assume that R is an orbital relation. This means that for any elements x, y, and z in the set, if xRy and yRz, then zRx.
Now, let's focus on the reflexive property. A relation is reflexive if every element is related to itself. In other words, for every element x in the set, xRx should hold.
To prove reflexivity, we pick an arbitrary element x in the set. Since R is orbital, we know that xRx holds if we can find an element y such that xRy and yRx.
Notice that if we choose y = x, then xRx is automatically satisfied. This is because if we substitute y = x into the orbital property, we have:
xRx (since xRx implies xRx).
Therefore, we have shown that if R is orbital, then R is reflexive.
Next, we move on to proving the second implication: "If R is both reflexive and orbital, then R is an equivalence relation." This means we need to show that if R is reflexive and orbital, then it must also be symmetric and transitive.
To prove symmetry, we need to show that if xRy, then yRx for any elements x and y in the set.
Since R is reflexive, we know that xRx holds for any element x. So, if we choose x = y, then xRy and yRx both hold.
Therefore, R is reflexive and symmetric.
Finally, to prove transitivity, we need to show that if xRy and yRz, then xRz for any elements x, y, and z in the set.
Since R is orbital, we know that if xRy and yRz, then zRx. We also know that R is reflexive, so xRx holds for any element x. By substituting z for x, we have:
zRx (since xRx implies zRx).
Therefore, R is reflexive, symmetric, and transitive.
By proving both the implications, we have concluded that R is an equivalence relation if and only if R is both reflexive and orbital.
first, do you know the definition of equivalence relation?
You don't have to prove that R is a relation -- they told you that.
So now use the properties of an equivalence relation.