0
$\begingroup$

I need help with this problem: Let $A$ resp. $B$ be a set, endowed with an equivalence relation $\sim_A$ resp. $\sim_B$. Defne a relation $\sim$ on $A \times B$ by setting

$$(a_1, b_1) \sim (a_2, b_2) \Leftrightarrow a_1\sim_A a_2 \ \text{and} \ b_1 \sim_B b_2.$$

Use the universal property for quotients to establish that there are functions

$$(A \times B)/\sim \ \rightarrow \ A/\sim_A$$ and $$(A \times B)/\sim \ \rightarrow \ B/\sim_B.$$

$\endgroup$
  • $\begingroup$ $A\times B\to A/\sim_A$ and $A\times B\to B/\sim_B$ are constant on equivalence clases, hence ... $\endgroup$ – Hagen von Eitzen Feb 26 '13 at 21:41
  • $\begingroup$ they are contained in the class? $\endgroup$ – Username Unknown Feb 26 '13 at 21:45
0
$\begingroup$

The kernel of a function $f:X\to Y$ is the equivalence relation $\theta_f$ on $X$ which satisfies $$x\,\theta_f\, x' \iff f(x)=f(x') \,.$$ Prove that $f$ factors through $X/R$ for some equivalence relation $R$ iff $R$ implies $\theta_f$ (i.e. $\forall x,x':\, xRx' \Rightarrow x\,\theta_f\, x'$).

Now use the composition $p:A\times B\to A\to A/\sim_A$ and check that $\sim$ implies $\theta_p$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.