1. ## Equivalence relation

Please could anyone confirm the following:

This concerns the relation as defined below on the complex numbers

z~w if Re z = Re w

This relation is reflexive, since the Re z = Re w, so z~w.

Let z,w be an element of a complex number, and suppose z~w. Then Re z = Re w. Hence Re w = Re z, so w~z. Thus the relation is symmetric.

Let z,w,v be an element of a complex number, and suppose that z~w and w~v. Then Re z = Re w and Re w = Re v. It follows that Re z = re v, so z~v. Thus the relation is transitive.

Hence this relation is an equivalence relation.

Thanks

2. Originally Posted by Arron
Please could anyone confirm the following
This concerns the relation as defined below on the complex numbers
z~w if Re z = Re w
Hence this relation is an equivalence relation.
That is correct.

3. Thanks alot.

4. Originally Posted by Arron
Please could anyone confirm the following:

This concerns the relation as defined below on the complex numbers

z~w if Re z = Re w

This relation is reflexive, since the Re z = Re w, so z~w.

Let z,w be an element of a complex number, and suppose z~w. Then Re z = Re w. Hence Re w = Re z, so w~z. Thus the relation is symmetric.

Let z,w,v be an element of a complex number, and suppose that z~w and w~v. Then Re z = Re w and Re w = Re v. It follows that Re z = re v, so z~v. Thus the relation is transitive.

Hence this relation is an equivalence relation.

Thanks

Reflexivity means that z~z for all the elements z, and not what you wrote. The rest is fine.

Tonio