3.7 Projecting to the quotient [1Z5]
[23M]Let \(A\) be a set and \(\sim \) an equivalence relation. We denote by
the quotient space, that is, the set of all equivalence classes; the canonical projection is the map \(\pi :A\to \frac A\sim \) that associates each \(x\in A\) with the class \([x]\in A/\sim \).
Suppose that the function \(f:A× A→ B\) is invariant for the equivalence relation \(∼\) in all its variables, i.e.
\[ ∀ x,y,v,w∈ A, \quad x ∼ y∧ v∼ w⇒ f(x,v)=f(y,w)\quad ; \]let \(\tilde f\) be the projection to the quotient \(\widetilde f:\frac A∼× \frac A∼→ B\) that satisfies
\[ f(x,y)=\widetilde f(𝜋(x),𝜋(y))\quad . \]If \(f\) is commutative (resp. associative) then \(\widetilde f\) is commutative (resp. associative).
If \(R\) is a relation in \(A× A\) invariant for \(∼\), and \(R\) is reflexive (resp symmetrical, antisymmetric, transitive) then \(\widetilde R\) is reflexive (resp symmetrical, antisymmetric, transitive).
If \(A\) and \(B\) are ordered and the order is invariant, and \(f\) is monotonic, then \(\widetilde f\) is monotonic.
[1Z7] (Replaces 06G) (Replaces 06H) Consider \(R\) a transitive and reflexive relation in \(A× A\); such a relation is called a preorder [ 43 ] ; we define \(x ∼ y\iff (xRy ∧ yRx)\) then \(∼\) is an equivalence relation, \(R\) is invariant for \(∼\), and \(\widetilde R\) (defined as in 81) is an order relation.
\(∼\) is clearly reflexive and symmetrical; is transitive because if \(x∼ y,y∼ z\) then \(xRy∧ yRx∧ yRz∧ zRy\) but being \(R\) transitive you get \(xRz∧ zRx\) i.e. \(x∼ z\)
Let \(x,y,\tilde x,\tilde y∈ X\) be such that \(x∼ \tilde x,y∼ \tilde y\) then we have \(xR\tilde x ∧ \tilde xRx∧ yR\tilde y∧ \tilde yRy\) if we add \(xRy\), by transitivity we get \(\tilde xR\tilde y\); and symmetrically.
Finally, we see that \({\widetilde R}\) is an order relation on \(Y\). Using the (well posed) definition ”\([x]{\widetilde R}[y] \iff xRy\)” we deduce that \({\widetilde R}\) is reflexive and transitive (as indeed stated in the previous proposition). \({\widetilde R}\) is also antisymmetric because if for \(z,w∈ A/∼\) you have \(z{\widetilde R}w∧ w{\widetilde R}z\) then taken \(x∈ z,y∈ w\) we have \(xRy ∧ yRx\) which means \(x∼ y\) and therefore \(z=w\).
- E82
[1Z8] \(ℤ\) are the relative integers with the usual operations. Let \(p≥ 1\) a fixed integer. Consider the equivalence relation
\[ n∼ m \iff p | (n-m) \]that is, they are equivalent when \(n-m\) is divisible by \(p\).
Show that there are \(p\) equivalence classes \([0],[1],\ldots [p-1]\) We indicate the quotient space with \(ℤ/(pℤ)\) or more briefly \(ℤ_ p\).
Show that the usual operations of sum, subtraction, product in \(ℤ\) pass to the quotient.