Chapter 12

Let \(\left\\{A_{i}: i \in I\right\\}\) be a partition of \(A .\) Let \(\left\\{B_{j}: j \in J\right\\}\) be a partition of \(B\). Prove that \(\left\\{A_{i} \times B_{j}:(i, j) \in I \times J\right\\}\) is a partition of \(A \times B\)

Let \(G\) be a group. In each of the following, a relation on \(G\) is defined. Prove it is an equivalence relation. Then describe the equivalence class of e. 1 If \(H\) is a subgroup of \(G\), let \(a \sim b\) ift \(a b^{-1} \in H\).

For each \(r \in \mathbb{R}, A_{r}=\\{(x, y): y=2 x+r\\}\)

For each integer \(r \in\\{0,1,2,3,4\\}\), let \(A_{r}\) be the set of all the integers which leave a remainder of \(r\) when divided by 5 . (That is, \(x \in A_{r}\) iff \(x=5 q+r\) for some integer \(q\).) Prove: \(\left\\{A_{0}, A_{1}, A_{2}, A_{3}, A_{4}\right\\}\) is a partition of \(\mathbb{Z}\)

For each \(r \in \mathbb{R}, A_{r}=\left\\{(x, y): x^{2}+y^{2}=r^{2}\right\\}\)

For each integer \(n\), let \(A_{n}=\\{x \in \mathbb{Q}: n \leqslant x

Let \(f: A \rightarrow B\) be a function. Define \(\sim\) by: \(a \sim b\) iff \(f(a)=f(b)\). Prove that \(\sim\) is an equivalence relation on \(A\). Describe its equivalence classes.

Let \([x\rceil\) denote the greatest integer \(\leqslant x\). In \(\mathbb{R}\), let \(a \sim b\) iff \(\lceil a\rceil=\lceil b\rceil\).

For each rational number \(r\), let \(A_{r}=\\{(m, n) \in \mathbb{Z} \times \mathbb{Z}: m / n=r\\} .\) Prove that \(\left\\{A_{r}: r \in \mathbb{Q}\right\\}\) is a partition of \(\mathbb{Z} \times \mathbb{Z}\)

Let \(f: A \rightarrow B\) be a surjective function, and let \(\left\\{B_{i}: i \in I\right\\}\) be a partition of \(B\). Prove that \(\left\\{f^{-1}\left(B_{i}\right): i \in I\right\\}\) is a partition of \(A .\) If \(\sim_{I}\) is the equivalence relation corresponding to the partition of \(B\), describe the equivalence relation corresponding to the partition of \(A .\) [REMARK: For any \(C \subseteq B, f^{-1}(C)=\\{x \in A: f(x) \in C\\}\).]

For \(r \in\\{0,1,2, \ldots, 9\\}\), let \(A_{r}\) be the set of all the integers whose units digit (in decimal notation) is equal to \(r .\) Prove: \(\left\\{A_{0}, A_{1}, A_{2}, \ldots, A_{9}\right\\}\) is a partition of \(\mathbb{Z} .\)

Let \(\sim_{1}\) and \(\sim_{2}\) be distinct equivalence relations on \(A\). Define \(\sim_{3}\) by: \(a \sim_{3} b\) iff \(a \sim_{1} b\) and \(a \sim_{2} b\). Prove that \(\sim_{3}\) is an equivalence relation on \(A\). If \([x]_{i}\) denotes the equivalence class of \(x\) for \(\sim_{i}(i=1,2,3)\), prove that \([x]_{3}=[x]_{1} \cap[x]_{2} .\)

For any rational number \(x\), we can write \(x=q+n / m\) where \(q\) is an integer and \(0 \leqslant n / m<1\). Call \(n / m\) the fractional part of \(x .\) For each rational \(r \in\\{x: 0 \leqslant x<1\\}\) let \(A_{r}=\\{x \in Q ;\) the fractional part of \(x\) is equal to \(r\\} .\) Prove: \(\left\\{A_{r}: 0 \leqslant r<1\right\\}\) is a partition of \(\mathbb{Q}\).

(x, y) \sim(u, v)\( iff \)x^{2}-y=u^{2}-v$

For each \(r \in \mathbb{R}\), let \(A_{r}=\\{(x, y) \in \mathbb{R} \times \mathbb{R}: x-y=r\\} .\) Prove: \(\left\\{A_{r}: r \in \mathbb{R}\right\\}\) is a partition of \(\mathbb{R} \times \mathbf{R}\)