Relations and Functions

$A \times B \neq B \times A$;

$A \times (B \times C) \neq (A \times B) \times C$.

$[a, b] \times [c, d]$, where $[a, b]$ and $[c, d]$ are segments of the real line $\mathbb {R}$;

$[a, b]^{2}$;

$[a, b]^{3}$;

$\mathbb {R}^{n}$.

$A \subseteq B$ and $C \subseteq D \Leftrightarrow A \times C \subseteq B \times D$;

$A = B$ and $C = D \Leftrightarrow A \times C = B \times D$.

$(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)$;

$\bigcap_{i \in I} A_{i} \times \bigcap_{i \in I} B_{i} = \bigcap_{i \in I} (A_{i} \times B_{i})$.

$(A \cup B) \times C = (A \times C) \cup (B \times C)$;

$A \times (B \cup C) = (A \times B) \cup (A \times C)$;

$(A \cup B) \times (C \cup D) = (A \times C) \cup (B \times C) \cup (A \times D) \cup (B \times D)$;

$(A \setminus B) \times C = (A \times C) \setminus (B \times C)$;

$A \times (B \setminus C) = (A \times B) \setminus (A \times C)$;

$A \times B = (A \times D) \cap (C \times B)$, where $A \subseteq C$ and $B \subseteq D$;

$U^{2} \setminus (A \times B) = [(U \setminus A) \times U] \cup [U \times (U \setminus B)]$;

$\bigcup_{k \in K} A_{k} \times \bigcup_{t \in T} B_{t} = \bigcup_{\langle k, t \rangle \in K \times T} (A_{k} \times B_{t})$;

$\bigcap_{k \in K} A_{k} \times \bigcap_{t \in T} B_{t} = \bigcap_{\langle k, t \rangle \in K \times T} (A_{k} \times B_{t})$.

$R = \left\{ \langle x, y \rangle \mid x, y \in \mathbb {N} \text{ and } x \text{ divides } y\right\}$;

$R = \left\{ \langle x, y \rangle \mid x, y \in \mathbb {N} \text{ and } y \text{ divides } x\right\}$;

$R = \left\{ \langle x, y \rangle \mid x, y \in \mathbb {R} \text{ and } x + y \leq 0\right\}$;

$R = \left\{ \langle x, y \rangle \mid x, y \in \mathbb {R} \text{ and } 2x \geq 3y\right\}$;

$R = \left\{ \langle x, y \rangle \mid x, y \in \left[-\frac{\pi }{2}, \frac{\pi }{2}\right] \text{ and } y \geq \sin x\right\}$.

$\delta R = \emptyset \Leftrightarrow R = \emptyset \Leftrightarrow \rho R = \emptyset$;

$\delta R^{-1} = \rho R$, $\rho R^{-1} = \delta R$;

$\delta_{R_{1} \cdot R_{2}} = R_{1}^{-1}(\rho R_{1} \cap \delta R_{2})$;

$\rho_{R_{1} \cdot R_{2}} = R_{2}(\rho R_{1} \cap \delta R_{2})$.

if $B \neq \emptyset$, then $\delta_{A \times B} = A$;

if $A \neq \emptyset$, then $\rho_{A \times B} = B$.

$R \cup R = R \cap R = R$;

$(R^{-1})^{-1} = R$;

$(R_{1} \cup R_{2})^{-1} = R_{1}^{-1} \cup R_{2}^{-1}$;

$(R_{1} \cap R_{2})^{-1} = R_{1}^{-1} \cap R_{2}^{-1}$;

$-R^{-1} = (-R)^{-1}$;

$\left(\bigcup_{i \in I} R_{i}\right)^{-1} = \bigcup_{i \in I} R_{i}^{-1}$;

$\left(\bigcap_{i \in I} R_{i}\right)^{-1} = \bigcap_{i \in I} R_{i}^{-1}$.

How many binary relations are there between the elements of the sets $A$ and $B$?

How many functions are there from $A$ to $B$?

How many 1--1 functions are there from $A$ to $B$?

For which $m$ and $n$ does there exist a one-to-one correspondence between $A$ and $B$?

$R_{1} \cdot (R_{2} \cdot R_{3}) = (R_{1} \cdot R_{2}) \cdot R_{3}$;

$(R_{1} \cdot R_{2})^{-1} = R_{2}^{-1} \cdot R_{1}^{-1}$;

$\left(\bigcup_{i \in I} R_{i}\right) \cdot Q = \bigcup_{i \in I} (R_{i} \cdot Q)$;

$Q \cdot \left(\bigcup_{i \in I} R_{i}\right) = \bigcup_{i \in I} (Q \cdot R_{i})$.

$Q \cdot \left(\bigcap_{i \in I} R_{i}\right) \subseteq \bigcap_{i \in I} (Q \cdot R_{i})$;

$\left(\bigcap_{i \in I} R_{i}\right) \cdot Q \subseteq \bigcap_{i \in I} (R_{i} \cdot Q)$;

in assertions (a) and (b) the inclusions cannot be replaced by equalities.

$Q \cdot R_{1} \subseteq Q \cdot R_{2}$;

$R_{1} \cdot Q \subseteq R_{2} \cdot Q$;

$R_{1}^{-1} \subseteq R_{2}^{-1}$.

if $B \neq \emptyset$, then $B^{A} \neq \emptyset$;

$B^{A} \subseteq P(A \times B)$.

$f^{-1}$ a function;

$f \cdot g$ a 1--1 function?

between $A \times B$ and $A_{1} \times B_{1}$;

between $A^{B}$ and $A_{1}^{B_{1}}$;

between $A \cup B$ and $A_{1} \cup B_{1}$, if $A \cap B = \emptyset$ and $A_{1} \cap B_{1} = \emptyset$.

$A \times B$ and $B \times A$;

$A \times (B \times C)$ and $(A \times B) \times C$;

$(A \times B)^{C}$ and $A^{C} \times B^{C}$;

$(A^{B})^{C}$ and $A^{B \times C}$;

$A^{B \cup C}$ and $A^{B} \times A^{C}$, if $B \cap C = \emptyset$;

$\prod_{i \in I} A_{i}$ and $\prod_{i \in I} A_{\varphi (i)}$, where $\varphi$ is a permutation of the set $I$;

$\prod_{i \in I} A_{i}$ and $\prod_{k \in K} \left(\prod_{j \in T_{k}} A_{j}\right)$, where $\bigcup_{k \in K} T_{k} = I$ and all the $T_{k}$ are pairwise disjoint;

$A^{I}$ and $\prod_{k \in K} A^{T_{k}}$, where $\bigcup_{k \in K} T_{k} = I$ and all the $T_{k}$ are pairwise disjoint.

$\varphi^{-1}$ is a one-to-one correspondence between $B$ and $A$;

$\varphi^{-1} \cdot \varphi = i_{B}$;

$\varphi \cdot \varphi^{-1} = i_{A}$.

$f(A \cup B) = f(A) \cup f(B)$;

$f\left(\bigcup_{i \in I} A_{i}\right) = \bigcup_{i \in I} f(A_{i})$.

$f(A \cap B) \subseteq f(A) \cap f(B)$;

$f\left(\bigcap_{i \in I} A_{i}\right) \subseteq \bigcap_{i \in I} f(A_{i})$,

$f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$;

$f^{-1}\left(\bigcup_{i \in I} A_{i}\right) = \bigcup_{i \in I} f^{-1}(A_{i})$;

$f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$;

$f^{-1}\left(\bigcap_{i \in I} A_{i}\right) = \bigcap_{i \in I} f^{-1}(A_{i})$;

$f^{-1}(A \setminus B) = f^{-1}(A) \setminus f^{-1}(B)$.

$A \subseteq f^{-1}(f(A))$;

$f(f^{-1}(B)) = B$;

$f(A) \cap B = f(A \cap f^{-1}(B))$;

$f(A) \cap B = \emptyset \Leftrightarrow A \cap f^{-1}(B) = \emptyset$;

$f(A) \subseteq B \Leftrightarrow A \subseteq f^{-1}(B)$.

$f^{*}(X \cap Y) = f^{*}(X) \cap f^{*}(Y)$;

$(f \cdot g)^{*}(X) = f^{*}(g^{*}(X))$.

$\chi_{U}^{U}(x) = 0$;

$\chi_{\emptyset }^{U}(x) = 1$;

$\chi_{U \setminus A}^{U}(x) = 1 - \chi_{A}^{U}(x)$;

$\chi_{A \cup B}^{U}(x) = \chi_{A}^{U}(x) \cdot \chi_{B}^{U}(x)$;

$\chi_{A \cap B}^{U}(x) = \chi_{A}^{U}(x) + \chi_{B}^{U}(x) - \chi_{A}^{U}(x) \cdot \chi_{B}^{U}(x)$;

$\chi_{A \setminus B}^{U}(x) = 1 - \chi_{B}^{U}(x) + \chi_{A \cup B}^{U}(x)$;

if $A = \bigcup_{i \in I} A_{i}$, then $\chi_{A}^{U}(x) = \min_{i \in I} \chi_{A_{i}}^{U}(x)$;

if $A = \bigcap_{i \in I} A_{i}$, then $\chi_{A}^{U}(x) = \max_{i \in I} \chi_{A_{i}}^{U}(x)$.

$\bigcup_{i \in I} \bigcap_{j \in J} A_{ij} = \bigcap_{f \in J^{I}} \bigcup_{i \in I} A_{i f(i)}$;

$\bigcap_{i \in I} \bigcup_{j \in J} A_{ij} = \bigcup_{f \in J^{I}} \bigcap_{i \in I} A_{i f(i)}$.

$\prod_{i \in I} A_{i} = \bigcap_{i \in I} \prod_{j \in J} A_{ij}$, where $A_{ii} = A_{i}$, $A_{ij} = X_{j}$ for $i \neq j$;

$\prod_{i \in I} X_{i} \setminus \prod_{i \in I} A_{i} = \bigcup_{i \in I} \prod_{j \in J} B_{ij}$, where $B_{ii} = X_{i} \setminus A_{i}$, $B_{ij} = X_{j}$ for $i \neq j$.

$\bigcap_{k \in K} \prod_{t \in T} A_{kt} = \prod_{t \in T} \bigcap_{k \in K} A_{kt}$;

if $A_{t_{1}} \cap A_{t_{2}} = \emptyset$ for $t_{1} \neq t_{2}$, then one can establish a one-to-one correspondence between $B^{\left(\bigcup_{t \in T} A_{t}\right)}$ and $\prod_{t \in T} B^{A_{t}}$;

one can establish a one-to-one correspondence between $\left(\prod_{t \in T} B_{t}\right)^{A}$ and $\prod_{t \in T} B_{t}^{A}$.