Binary relations

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Problem: \( A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \cdot B, P_{2} \subseteq B^{2} \). Represent \( P_{1}, P_{2} \) graphically. Find the matrix \( \left(P_{1} \circ P_{2}\right)^{-1} \). Using the matrix, check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive? What class of relations does it belong to? \[ \begin{array}{l} P_{1}=\{(a, 1),(a, 2),(a, 3),(a, 4),(b, 3),(c, 2)\} \\ P_{2}=\{(1,1),(1,4),(2,2),(2,3),(3,3),(3,2),(4,1),(4,4)\} . \end{array} \]

6.2.1 Binary relations

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Problem: Find the domain of definition, the domain of values of the relation \( P \). Is the relation \( P \) reflexive, antireflexive, symmetrical, antisymmetric, nonsymmetrical, transitive? What class of relations does it belong to? Justify your answer. \[ P \subseteq R^{2},(x, y) \in P \Leftrightarrow x \cdot y>1 . \]

6.2.2 Binary relations

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Problem: Present the given relationship in the form of a graph and a matrix. Determine the properties of the relation (symmetry, reflexivity, transitivity). \[ R=\left\{\begin{array}{l} a b \\ a c \\ c b \\ a a \end{array}\right\} . \]

6.2.3 Binary relations

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Problem: Show that \( R \circ R \subseteq R \) for any transitive binary relation \( R \). Can this inclusion be strict?

6.2.4 Binary relations

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Problem: Let's consider on the set \( \mathbb{Z}^{2} \) the binary relation \( (k, l) \sim(m, n) \), which means that \( m+n-k-l \) is divisible by 3 . Is it an equivalence? Draw on checkered paper all such points \( (m, n), 0 \leq \) \( m, n \leq 10 \) that \( (m, n) \sim(0,0) \). Find the maximum number of pairwise incomparable points.

6.2.5 Binary relations

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Problem: Show that the relation \( x \equiv y(\bmod \mathbb{Z}) \), meaning that \( x-y \in \mathbb{Z} \), gives an equivalence on the set of real numbers \( R \), and construct an explicit bijection between the quotient set \( R / \mathbb{Z} \) and the unit circle \( S^{1} \stackrel{\text { def }}{=}\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} \) in \( \mathbb{R}^{2} \). Does the standard order in \( \mathbb{R} \) induce any order in \( \mathbb{R} / \mathbb{Z} \) ? Construct an explicit bijection between the functions \( S^{1} \rightarrow \mathbb{R} \) and the periodic functions \( \mathbb{R} \rightarrow \mathbb{R} \) of period 1 .

6.2.6 Binary relations

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Problem: A binary relation is given: \( R=\{(x, y) \mid x, y \in \mathbb{N} \wedge \) \( y: x\} \). Find: \( \delta_{R}, \rho_{R}, R^{1}, R \circ R^{-1}, R \circ R \).

6.2.7 Binary relations

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Problem: A binary relation is given: \( R=\{(x, y) \mid x, y \in \mathbb{N} \wedge \) \( x: y\} \). Find: \( \delta_{R}, \rho_{R}, R^{1}, R \circ R^{-1}, R \circ R \).

6.2.8 Binary relations

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Problem: A binary relation is given: \( R=\{(x, y) \mid x, y \in D \wedge \) \( y+x \leq 0\} \). Find: \( \delta_{R}, \rho_{R}, R^{1}, R \circ R^{-1}, R \circ R \).

6.2.9 Binary relations

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Problem: A binary relation is given: \( R=\{(x, y) \mid x, y \in D \wedge \) \( 2 x \geq 3 y\} \). Find: \( \delta_{R}, \rho_{R}, R^{1}, R \circ R^{-1}, R \circ R \).

6.2.10 Binary relations

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Problem: A binary relation is given: \( R=\{(x, y) \mid x, y \in \) \( \left.\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \wedge y \geq \sin x\right\} \). Find: \( \delta_{R}, \rho_{R}, R^{1}, R \circ R^{-1}, R \circ R \).

6.2.11 Binary relations

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Problem: The binary relation \( P \) is given; find its domain and range. Check by the definition whether relation \( P \) is reflexive, symmetric, antisymmetric, transitive. \[ P \subseteq \mathbb{Z}^{2}, P=\{(x, y) \mid x=-y\} . \]

6.2.12 Binary relations

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Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations \( P_{1} \subseteq A \times B, P_{2} \subseteq B^{2} \). Plot \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domain and range of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 3),(b, 4),(b, 3),(c, 1),(c, 2),(c, 4)\} ; \\ P_{2}=\{(1,2),(1,3),(1,4),(2,3),(4,3),(4,2)\} . \\ A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \times B, P_{2} \subseteq B^{2} . \end{array} \]

6.2.13 Binary relations

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Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations \( P_{1} \subseteq A \times B ; P_{2} \subseteq B^{2} \). Plot \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domain and the range of all three relation: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 2),(a, 4),(b, 1),(b, 2),(b, 4),(c, 2),(c, 4)\}, \\ P_{2}=\{(1,1),(2,2),(2,4),(3,3),(3,2),(4,4),(1,3),(4,1)\} . \end{array} \]

6.2.14 Binary relations

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Problem: The binary relation \( P \); find its domain and range. Check by the definition whether the relation \( P \) is reflexive, symmetric, antisymmetric, transitive. \( P \subseteq \) \( \subseteq Z^{2}, P=\{(x, y) \mid(x+2 \cdot y) \) multiple of 2\( \} \).

6.2.15 Binary relations

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